signal and linear system analysis and linear system analysis contents 2.1 signal models ... a signal...
TRANSCRIPT
Chapter 2Signal and Linear System Analysis
Contents
2.1 Signal Models . . . . . . . . . . . . . . . . . . . . . 2-32.1.1 Deterministic and Random Signals . . . . . . 2-32.1.2 Periodic and Aperiodic Signals . . . . . . . . 2-32.1.3 Phasor Signals and Spectra . . . . . . . . . . 2-42.1.4 Singularity Functions . . . . . . . . . . . . . 2-7
2.2 Signal Classifications . . . . . . . . . . . . . . . . . 2-112.3 Generalized Fourier Series . . . . . . . . . . . . . . 2-142.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . 2-20
2.4.1 Complex Exponential Fourier Series . . . . . 2-202.4.2 Symmetry Properties of the Fourier Coeffi-
cients . . . . . . . . . . . . . . . . . . . . . 2-232.4.3 Trigonometric Form . . . . . . . . . . . . . 2-252.4.4 Parseval’s Theorem . . . . . . . . . . . . . . 2-262.4.5 Line Spectra . . . . . . . . . . . . . . . . . 2-262.4.6 Numerical Calculation of Xn . . . . . . . . . 2-302.4.7 Other Fourier Series Properties . . . . . . . . 2-37
2.5 Fourier Transform . . . . . . . . . . . . . . . . . . . 2-38
2-1
CONTENTS
2.5.1 Amplitude and Phase Spectra . . . . . . . . 2-392.5.2 Symmetry Properties . . . . . . . . . . . . . 2-392.5.3 Energy Spectral Density . . . . . . . . . . . 2-402.5.4 Transform Theorems . . . . . . . . . . . . . 2-422.5.5 Fourier Transforms in the Limit . . . . . . . 2-512.5.6 Fourier Transforms of Periodic Signals . . . 2-532.5.7 Poisson Sum Formula . . . . . . . . . . . . 2-59
2.6 Power Spectral Density and Correlation . . . . . . . 2-602.6.1 The Time Average Autocorrelation Function 2-612.6.2 Power Signal Case . . . . . . . . . . . . . . 2-622.6.3 Properties of R.�/ . . . . . . . . . . . . . . 2-63
2.7 Linear Time Invariant (LTI) Systems . . . . . . . . . 2-742.7.1 Stability . . . . . . . . . . . . . . . . . . . . 2-762.7.2 Transfer Function . . . . . . . . . . . . . . . 2-762.7.3 Causality . . . . . . . . . . . . . . . . . . . 2-772.7.4 Properties of H.f / . . . . . . . . . . . . . . 2-782.7.5 Input/Output with Spectral Densities . . . . . 2-822.7.6 Response to Periodic Inputs . . . . . . . . . 2-822.7.7 Distortionless Transmission . . . . . . . . . 2-832.7.8 Group and Phase Delay . . . . . . . . . . . . 2-842.7.9 Nonlinear Distortion . . . . . . . . . . . . . 2-872.7.10 Ideal Filters . . . . . . . . . . . . . . . . . . 2-892.7.11 Realizable Filters . . . . . . . . . . . . . . . 2-912.7.12 Pulse Resolution, Risetime, and Bandwidth . 2-101
2.8 Sampling Theory . . . . . . . . . . . . . . . . . . . 2-1062.9 The Hilbert Transform . . . . . . . . . . . . . . . . 2-1062.10 The Discrete Fourier Transform and FFT . . . . . . . 2-106
2-2 ECE 5625 Communication Systems I
2.1. SIGNAL MODELS
2.1 Signal Models
2.1.1 Deterministic and Random Signals
� Deterministic Signals, used for this course, can be modeled ascompletely specified functions of time, e.g.,
x.t/ D A.t/ cosŒ2�f0.t/t C �.t/�
– Note that here we have also made the amplitude, fre-quency, and phase functions of time
– To be deterministic each of these functions must be com-pletely specified functions of time
� Random Signals, used extensively in Comm Systems II, takeon random values with known probability characteristics, e.g.,
x.t/ D x.t; �i/
where �i corresponds to an elementary outcome from a samplespace in probability theory
– The �i create a ensemble of sample functions x.t; �i/, de-pending upon the outcome drawn from the sample space
2.1.2 Periodic and Aperiodic Signals
� A deterministic signal is periodic if we can write
x.t C nT0/ D x.t/
for any integer n, with T0 being the signal fundamental period
ECE 5625 Communication Systems I 2-3
CONTENTS
� A signal is aperiodic otherwise, e.g.,
….t/ D
(1; jt j � 1=2
0; otherwise
(a) periodic signal, (b) aperiodic signal, (c) random signal
2.1.3 Phasor Signals and Spectra
� A complex sinusoid can be viewed as a rotating phasor
Qx.t/ D Aej.!0tC�/; �1 < t <1
� This signal has three parameters, amplitude A, frequency !0,and phase �
� The fixed phasor portion is Aej� while the rotating portion isej!0t
2-4 ECE 5625 Communication Systems I
2.1. SIGNAL MODELS
� This signal is periodic with period T0 D 2�=!0
� It also related to the real sinusoid signal A cos.!0t C �/ viaEuler’s theorem
x.t/ D Re˚Qx.t/
D Re
˚A cos.!0t C �/C jA sin.!0t C �/
D A cos.!0t C �/
(a) obtain x.t/ from Qx.t/, (b) obtain x.t/ from Qx.t/ and Qx�.t/
� We can also turn this around using the inverse Euler formula
x.t/ D A cos.!0t C �/
D1
2Qx.t/C
1
2Qx�.t/
DAej.!0tC�/ C Ae�j.!0tC�/
2
� The frequency spectra of a real sinusoid is the line spectra plot-ted in terms of the amplitude and phase versus frequency
ECE 5625 Communication Systems I 2-5
CONTENTS
� The relevant parameters are A and � for a particular f0 D!0=.2�/
(a) Single-sided line spectra, (b) Double-sided line spectra
� Both the single-sided and double-sided line spectra, shownabove, correspond to the real signal x.t/ D A cos.2�f0t C �/
Example 2.1: Multiple Sinusoids
� Suppose that
x.t/ D 4 cos.2�.10/t C �=3/C 24 sin.2�.100/t � �=8/
� Find the two-sided amplitude and phase line spectra of x.t/
� First recall that cos.!0t � �=2/ D sin.!0t /, so
x.t/ D 4 cos.2�.10/t C �=3/C 24 cos.2�.100/t � 5�=8/
� The complex sinusoid form is directly related to the two-sidedline spectra since each real sinusoid is composed of positiveand negative frequency complex sinusoids
x.t/ D 2hej.2�.10/tC�=3/ C e�j.2�.10/tC�=3/
iC 12
hej.2�.100/t�5�=8/ C e�j.2�.100/t�5�=8/
i2-6 ECE 5625 Communication Systems I
2.1. SIGNAL MODELS
f (Hz)
f (Hz)
Am
plitu
dePh
ase
10
12
2
100-100 -10
5π/8
-5π/8-π/3
π/3
Two-sided amplitude and phase line spectra
2.1.4 Singularity Functions
Unit Impulse (Delta) Function
� Singularity functions, such as the delta function and unit step
� The unit impulse function, ı.t/ has the operational propertiesZ t2
t1
ı.t � t0/ dt D 1; t1 < t0 < t2
ı.t � t0/ D 0; t ¤ t0
which implies that for x.t/ continuous at t D t0, the siftingproperty holds Z
1
�1
x.t/ı.t � t0/ dt D x.t0/
– Alternatively the unit impulse can be defined asZ1
�1
x.t/ı.t/ dt D x.0/
ECE 5625 Communication Systems I 2-7
CONTENTS
� Properties:
1. ı.at/ D ı.t/=jaj
2. ı.�t / D ı.t/
3. Sifting property special cases
Z t2
t1
x.t/ı.t � t0/ dt D
8̂̂<̂:̂x.t0/; t1 < t0 < t2
0; otherwise
undefined; t0 D t1 or t0 D t2
4. Sampling property
x.t/ı.t � t0/ D x.t0/ı.t � t0/
for x.t/ continuous at t D t05. Derivative propertyZ t2
t1
x.t/ı.n/.t � t0/ dt D .�1/nx.n/.t0/
D .�1/nd nx.t/
dtn
ˇ̌̌̌tDt0
Note: Dealing with the derivative of a delta function re-quires care
� A test function for the unit impulse function helps our intuitionand also helps in problem solving
� Two functions of interest are
ı�.t/ D1
2�…
�t
2�
�D
(12�; jt j � �
0; otherwise
ı1�.t/ D �
�1
�tsin�t
�
�22-8 ECE 5625 Communication Systems I
2.1. SIGNAL MODELS
Test functions for the unit impulse ı.t/: (a) ı�.t/, (b) ı1�.t/
� In both of the above test functions letting � ! 0 results in afunction having the properties of a true delta function
Unit Step Function
� The unit step function can be defined in terms of the unit im-pulse
u.t/ �
Z t
�1
ı.�/ d� D
8̂̂<̂:̂0; t < 0
1; t > 0
undefined; t D 0
alsoı.t/ D
du.t/
dt
ECE 5625 Communication Systems I 2-9
CONTENTS
Example 2.2: Unit Impulse 1st-Derivative
� Consider Z1
�1
x.t/ı0.t/ dt
� Using the rectangular pulse test function, ı�.t/, we note that
ı�.t/ D1
2�…
�t
2�
�alsoD
1
2�
�u.t C �/ � u.t � �/
�and
dı�.t/
dtD
1
2�
�ı.t C �/ � ı.t � �/
�� Placing the above in the integrand with x.t/ we obtain, with
the aid of the sifting property, thatZ1
�1
x.t/ı0.t/ dt D lim�!0
1
2�
�x.t C �/ � x.t � �/
�D lim
�!0
��x.t � �/ � x.t C �/
�2�
D �x0.0/
2-10 ECE 5625 Communication Systems I
2.2. SIGNAL CLASSIFICATIONS
2.2 Signal Classifications
� From circuits and systems we know that a real voltage or cur-rent waveform, e.t/ or i.t/ respectively, measured with respec-tive a real resistance R, the instantaneous power is
P.t/ D e.t/i.t/ D i2.t/R W
� On a per-ohm basis, we obtain
p.t/ D P.t/=R D i2.t/ W/ohm
� The average energy and power can be obtain by integratingover the interval jt j � T with T !1
E D limT!1
Z T
�T
i2.t/ dt Joules/ohm
P D limT!1
1
2T
Z T
�T
i2.t/ dt W/ohm
� In system engineering we take the above energy and powerdefinitions, and extend them to an arbitrary signal x.t/, pos-sibly complex, and define the normalized energy (e.g. 1 ohmsystem) as
E�D lim
T!1
Z T
�T
jx.t/j2 dt D
Z1
�1
jx.t/j2 dt
P�D lim
T!1
1
2T
Z T
�T
jx.t/j2 dt
ECE 5625 Communication Systems I 2-11
CONTENTS
� Signal Classes:
1. x.t/ is an energy signal if and only if 0 < E <1 so thatP D 0
2. x.t/ is a power signal if and only if 0 < P < 1 whichimplies that E !1
Example 2.3: Real Exponential
� Consider x.t/ D Ae�˛tu.t/ where ˛ is real
� For ˛ > 0 the energy is given by
E D
Z1
0
�Ae�˛t
�2dt D
A2e�2˛t
�2˛
ˇ̌̌̌1
0
DA2
2˛
� For ˛ D 0 we just have x.t/ D Au.t/ and E !1
� For ˛ < 0 we also have E !1
� In summary, we conclude that x.t/ is an energy signal for ˛ >0
� For ˛ > 0 the power is given by
P D limT!1
1
2T
A2
2˛
�1 � e�˛T
�D 0
� For ˛ D 0 we have
P D limT!1
1
2T� A2T D
A2
2
2-12 ECE 5625 Communication Systems I
2.2. SIGNAL CLASSIFICATIONS
� For ˛ < 0 we have P !1
� In summary, we conclude that x.t/ is a power signal for ˛ D 0
Example 2.4: Real Sinusoid
� Consider x.t/ D A cos.!0t C �/; �1 < t <1
� The signal energy is infinite since upon squaring, and integrat-ing over one cycle, T0 D 2�=!0, we obtain
E D limN!1
Z NT0=2
�NT0=2
A2 cos2.!0t C �/ dt
D limN!1
N
Z T0=2
�T0=2
A2 cos2.!0t C �/ dt
D limN!1
NA2
2
Z T0=2
�T0=2
�1C cos.2!0t C 2�/ dt
D limN!1
NA2
2� T0!1
� The signal average power is finite since the above integral isnormalized by 1=.NT0/, i.e.,
P D limN!1
1
NT0�N
A2
2� T0 D
A2
2
ECE 5625 Communication Systems I 2-13
CONTENTS
2.3 Generalized Fourier Series
The goal of generalized Fourier series is to obtain a representation ofa signal in terms of points in a signal space or abstract vector space.The coordinate vectors in this case are orthonomal functions. Thecomplex exponential Fourier series is a special case.
� Let EA be a vector in a three dimensional vector space
� Let Ea1; Ea2, and Ea3 be linearly independent vectors in the samethree dimensional space, then
c1Ea1 C c2Ea2 C c3Ea3 D 0 .zero vector/
only if the constants c1 D c2 D c3 D 0
� The vectors Ea1; Ea2, and Ea3 also span the three dimensionalspace, that is for any vector EA there exists a set of constantsc1; c2, and c3 such that
EA D c1Ea1 C c2Ea2 C c3Ea3
� The set fEa1; Ea2; Ea3g forms a basis for the three dimensionalspace
� Now let fEa1; Ea2; Ea3g form an orthogonal basis, which impliesthat
Eai � Eaj D .Eai ; Eaj / D hEai ; Eaj i D 0; i ¤ j
which says the basis vectors are mutually orthogonal
� From analytic geometry (and linear algebra), we know that wecan find a representation for EA as
EA D.Ea1 � EA/
jEa1j2C.Ea2 � EA/
jEa2j2C.Ea3 � EA/
jEa3j2
2-14 ECE 5625 Communication Systems I
2.3. GENERALIZED FOURIER SERIES
which implies that
EA D
3XiD1
ci Eai
where
ci DEai � EA
jEai j2; i D 1; 2; 3
is the component of EA in the Eai direction
� We now extend the above concepts to a set of orthogonal func-tions f�1.t/; �2.t/; : : : ; �N .t/g defined on to � t � t0 C T ,where the dot product (inner product) associated with the �n’sis
��m.t/; �n.t/
�D
Z t0CT
t0
�m.t/��
n.t/ dt
D cnımn D
(cn; n D m
0; n ¤ m
� The �n’s are thus orthogonal on the interval Œt0; t0 C T �
� Moving forward, let x.t/ be an arbitrary function on Œt0; t0CT �,and consider approximating x.t/ with a linear combination of�n’s, i.e.,
x.t/ ' xa.t/ D
NXnD1
Xn�n.t/; t0 � t � t0 C T;
where a denotes approximation
ECE 5625 Communication Systems I 2-15
CONTENTS
� A measure of the approximation error is the integral squarederror (ISE) defined as
�N D
ZT
ˇ̌x.t/ � xa.t/
ˇ̌2dt;
whereRT
denotes integration over any T long interval
� To find the Xn’s giving the minimum �N we expand the aboveintegral into three parts (see homework problems)
�N D
ZT
jx.t/j2 dt �
NXnD1
1
cn
ˇ̌̌̌ZT
x.t/��n.t/ dt
ˇ̌̌̌2C
NXnD1
cn
ˇ̌̌̌Xn �
1
cn
ZT
x.t/��n.t/ dt
ˇ̌̌̌2– Note that the first two terms are independent of the Xn’s
and the last term is nonnegative (missing steps are in texthomework problem 2.14)
� We conclude that �N is minimized for each n if each elementof the last term is made zero by setting
Xn D1
cn
ZT
x.t/��n.t/ dt Fourier Coefficient
� This also results in
��N�
min D
ZT
jx.t/j2 dt �
NXnD1
cnjXnj2
2-16 ECE 5625 Communication Systems I
2.3. GENERALIZED FOURIER SERIES
� Definition: The set of of �n’s is complete if
limN!1
.�N /min D 0
forRTjx.t/j2 dt <1
– Even if though the ISE is zero when using a completeset of orthonormal functions, there may be isolated pointswhere x.t/ � xa.t/ ¤ 0
� Summary
x.t/ D l.i.m.1XnD1
Xn�n.t/
Xn D1
cn
ZT
x.t/��n.t/ dt
– The notation l.i.m. stands for limit in the mean, which isa mathematical term referring to the fact that ISE is theconvergence criteria
� Parseval’s theorem: A consequence of completeness isZT
jx.t/j2 dt D
1XnD1
cnjXnj2
ECE 5625 Communication Systems I 2-17
CONTENTS
Example 2.5: A Three Term Expansion
� Approximate the signal x.t/ D cos 2�t on the interval Œ0; 1�using the following basis functions
0.2 0.4 0.6 0.8 1
-1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
0.2 0.4 0.6 0.8 1
-1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1x(t) φ
1(t)
t t
0.2 0.4 0.6 0.8 1
-1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
0.2 0.4 0.6 0.8 1
-1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
φ2(t) φ
3(t)
t t
Signal x.t/ and basis functions �i.t/; i D 1; 2; 3
� To begin with it should be clear that �1.t/; �2.t/, and �3.t/are mutually orthogonal since the integrand associated with theinner product, �i.t/ � ��j .t/ D 0, for i ¤ j; i; j D 1; 2; 3
2-18 ECE 5625 Communication Systems I
2.3. GENERALIZED FOURIER SERIES
� Before finding the Xn’s we need to find the cn’s
c1 D
ZT
j�1.t/j2 dt
Z 1=4
0
j1j2 dt D 1=4
c2 D
ZT
j�2.t/j2 dt D 1=2
c3 D
ZT
j�3.t/j2 dt D 1=4
� Now we can compute the Xn’s:
X1 D 4
ZT
x.t/��1 .t/ dt
D 4
Z 1=4
0
cos.2�t/ dt D2
�sin.2�t/
ˇ̌̌1=40D2
�
X2 D 2
Z 3=4
1=4
cos.2�t/ dt D1
�sin.2�t/
ˇ̌̌3=41=4D�2
�
X3 D 4
Z 1
3=4
cos.2�t/ dt D2
�sin.2�t/
ˇ̌̌13=4D2
�
t
x(t)
xa(t)
0.2 0.4 0.6 0.8 1
-1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
-2/π
2/π
Functional approximation
ECE 5625 Communication Systems I 2-19
CONTENTS
� The integral-squared error, �N , can be computed as follows:
�N D
ZT
ˇ̌̌̌x.t/ �
3XnD1
Xn�n.t/
ˇ̌̌̌2dt
D
ZT
jx.t/j2 dt �
3XnD1
cnjXnj2
D1
2�1
4
ˇ̌̌̌2
�
ˇ̌̌̌2�1
2
ˇ̌̌̌2
�
ˇ̌̌̌2�1
4
ˇ̌̌̌2
�
ˇ̌̌̌2D1
2�
ˇ̌̌̌2
�
ˇ̌̌̌2D 0:0947
2.4 Fourier Series
When we choose a particular set of basis functions we arrive at themore familiar Fourier series.
2.4.1 Complex Exponential Fourier Series
� A set of �n’s that is complete is
�n.t/ D ejn!0t ; n D 0;˙1;˙2; : : :
over the interval .t0; t0 C T0/ where !0 D 2�=T0 is the periodof the expansion interval
2-20 ECE 5625 Communication Systems I
2.4. FOURIER SERIES
proof of orthogonality��m.t/; �n.t/
�D
Z t0CT0
t0
ejm2�tT0 e
�jn2�tT0 dt D
Z t0CT0
t0
ej 2�T0
.m�n/tdt
D
8̂̂<̂:̂R t0CT0t0
dt; m D nR t0CT0t0
�cosŒ2�.m � n/t=T0�
Cj sinŒ2�.m � n/t=T0��dt; m ¤ n
D
(T0; m D n
0; m ¤ n
We also conclude that cn D T0
� Complex exponential Fourier series summary:
x.t/ D
1XnD�1
Xnejn!0t ; t0 � t � t0 C T0
where Xn D1
T0
ZT0
x.t/e�jn!0t
� The Fourier series expansion is unique
Example 2.6: x.t/ D cos2!0t
� If we expand x.t/ into complex exponentials we can immedi-ately determine the Fourier coefficients
x.t/ D1
2C1
2cos 2!0t
D1
2C1
4ej 2!0t C
1
4e�j 2!0t
ECE 5625 Communication Systems I 2-21
CONTENTS
� The above implies that
Xn D
8̂̂<̂:̂12; n D 014; n D ˙2
0; otherwise
Time Average Operator
� The time average of signal v.t/ is defined as
hv.t/i�D lim
T!1
1
2T
Z T
�T
v.t/ dt
� Note that
hav1.t/C bv2.t/i D ahv1.t/i C bhv2.t/i;
where a and b are arbitrary constants
� If v.t/ is periodic, with period T0, then
hv.t/i D1
T0
ZT0
v.t/ dt
� The Fourier coefficients can be viewed in terms of the timeaverage operator
� Let v.t/ D x.t/e�jn!0t using e�j� D cos � � j sin � , we findthat
Xn D hv.t/i D hx.t/e�jn!0ti
D hx.t/ cosn!0ti � j hx.t/ sinn!0ti
2-22 ECE 5625 Communication Systems I
2.4. FOURIER SERIES
2.4.2 Symmetry Properties of the Fourier Coef-ficients
� For x.t/ real, the following coefficient symmetry propertieshold:
1. X�n D X�n2. jXnj D jX�nj
3. †Xn D �†X�n
proof
X�n D
�1
T0
ZT0
x.t/e�jn!0t dt
��D
1
T0
ZT0
x.t/e�j.�n/!0t dt D X�n
since x�.t/ D x.t/
� Waveform symmetry conditions produce special results too
1. If x.�t / D x.t/ (even function), then
Xn D Re˚Xn; i.e., Im
˚XnD 0
2. If x.�t / D �x.t/ (odd function), then
Xn D Im˚Xn; i.e., Re
˚XnD 0
3. If x.t ˙ T0=2/ D �x.t/ (odd half-wave symmetry), then
Xn D 0 for n even
ECE 5625 Communication Systems I 2-23
CONTENTS
Example 2.7: Odd Half-wave Symmetry Proof
� Consider
Xn D1
T0
Z t0DT0=2
t0
x.t/e�jn!0t dtC1
T0
Z t0CT0
t0CT0=2
x.t 0/e�jn!0t0
dt 0
� In the second integral we change variables by letting t D t 0 �
T0=2
Xn D1
T0
Z t0CT0=2
t0
x.t/e�jn!0t dt
C1
T0
Z tCT0=2
t0
x.t � T0=2/„ ƒ‚ …�x.t/
e�jn!0.tCT0=2/ dt
D
�1 � e�jn!0T0=2
� 1T0
Z t0CT0=2
t0
x.t/e�jn!0t dt
but n!0.T0=2/ D n.2�=T0/.T0=2/ D n� , thus
1 � e�jn� D
(2; n odd
0; n even
� We thus see that the even indexed Fourier coefficients are in-deed zero under odd half-wave symmetry
2-24 ECE 5625 Communication Systems I
2.4. FOURIER SERIES
2.4.3 Trigonometric Form
� The complex exponential Fourier series can be arranged as fol-lows
x.t/ D
1XnD�1
Xnejn!0t
D X0 C
1XnD1
�Xne
jn!0t CX�ne�jn!0t
�� For real x.t/, we may know that jX�nj D jXnj and †Xn D�†X�n, so
x.t/ D X0 C
1XnD1
�jXnje
j Œn!0tC†Xn� C jXnje�j Œn!0tC†Xn�
�D X0 C 2
1XnD1
jXnj cos�n!0t C†Xn
�since cos.x/ D .ejx C e�jx/=2
� From the trig identity cos.uC v/ D cosu cos v� sinu sin v, itfollows that
x.t/ D X0 C
1XnD1
An cos.n!0t /C1XnD1
Bn sin.n!0t /
where
An D 2hx.t/ cos.n!0t /iBn D 2hx.t/ sin.n!0t /i
ECE 5625 Communication Systems I 2-25
CONTENTS
2.4.4 Parseval’s Theorem
� Fourier series analysis are generally used for periodic signals,i.e., x.t/ D x.t C nT0/ for any integer n
� With this in mind, Parseval’s theorem becomes
P D1
T0
ZT0
jx.t/j2 dt D
1XnD�1
jXnj2
D X20 C 2
1XnD1
jXnj2 .W/
Note: A 1 ohm system is assumed
2.4.5 Line Spectra
� Line spectra was briefly reviewed earlier for simple signals
� For any periodic signal having Fourier series representation wecan obtain both single-sided and double-sided line spectra
� The double-sided magnitude and phase line spectra is mosteasily obtained form the complex exponential Fourier series,while the single-sided magnitude and phase line spectra can beobtained from the trigonometric form
Double-sidedmag. and phase
()
1XnD�1
Xnej 2�.nf0/t
Single-sidedmag. and phase
() X0 C 2
1XnD1
jXnj cosŒ2�.nf0/t C†Xn�
2-26 ECE 5625 Communication Systems I
2.4. FOURIER SERIES
– For the double-sided simply plot as lines jXnj and †Xnversus nf0 for n D 0;˙1;˙2; : : :
– For the single-sided plot jX0j and †X0 as a special casefor n D 0 at nf0 D 0 and then plot 2jXnj and †X0 versusnf0 for n D 1; 2; : : :
Example 2.8: Cosine Squared
� Consider
x.t/ D A cos2.2�f0t C �/ DA
2CA
2cos
�2�.2f0/t C 2�1
�DA
2CA
4ej 2�1ej 2�.2f0/t C
A
4e�j 2�1e�j 2�.2f0/t
Double-Sided
Single-Sided
-2f0
-2f0
2f0
2f0
2f0
2f0
f
f f
f
ECE 5625 Communication Systems I 2-27
CONTENTS
Example 2.9: Pulse Train
t
A
x(t)
0 τ T0T
0 + τ-T
0-2T
0
. . .. . .
Periodic pulse train
� The pulse train signal is mathematically described by
x.t/ D
1XnD�1
A…
�t � nT0 � �=2
�
�
� The Fourier coefficients are
Xn D1
T0
Z �
0
Ae�j 2�.nf0/t dt DA
T0�e�j 2�.nf0/t
�j 2�.nf0/
ˇ̌̌�0
DA
T0�1 � e�j 2�.nf0/�
j 2�.nf0/
DA�
T0�ej�.nf0/� � e�j�.nf0/�
.2j /�.nf0/�� e�j�.nf0/�
DA�
T0�
sinŒ�.nf0/��Œ�.nf0/��
� e�j�.nf0/�
� To simplify further we define
sinc.x/ �D
sin.�x/�x
2-28 ECE 5625 Communication Systems I
2.4. FOURIER SERIES
� Finally,
Xn DA�
T0sinc.nf0�/e�j�.nf0/� ; n D 0;˙1;˙2; : : :
� To plot the line spectra we need to find jXnj and †Xn
jXnj DA�
T0jsincŒ.nfo/��j
†Xn D
8̂̂<̂:̂��.nf0/�; sinc.nfo�/ > 0
��.nf0/� C �; nf0 > 0 and sinc.nf0�/ < 0
��.nf0/� � �; nf0 < 0 and sinc.nf0�/ < 0
� Pulse train double-sided line spectra for � D 0:125 using Python,specifically use ssd.line_spectra()
� As a specific example enter the following at the IPython com-mand prompt
n = arange(0,25+1) # Get 0 through 25 harmonicstau = 0.125; f0 = 1; A = 1;Xn = A*tau*f0*sinc(n*f0*tau)*exp(-1j*pi*n*f0*tau)figure(figsize=(6,2))f = n # Assume a fundamental frequency of 1 Hz so f = nssd.line_spectra(f,Xn,mode=’mag’,fsize=(6,2))xlim([-25,25]);figure(figsize=(6,2))ssd.line_spectra(f,Xn,mode=’phase’,fsize=(6,2))xlim([-25,25]);
ECE 5625 Communication Systems I 2-29
CONTENTS
0 0.125fAt = 0 1, 0.125f t ==
1 8t =
Phase slope0.125
ff
p tp
-=- ´=
2.4.6 Numerical Calculation of Xn
� Here we consider a purely numerical calculation of the Xk co-efficients from a single period waveform description of x.t/
� In particular, we will use the numpy fast Fourier transform(FFT) function to carry out the numerical integration
� By definition
Xk D1
T0
ZT0
x.t/e�j 2�kf0t dt; k D 0;˙1;˙2; : : :
2-30 ECE 5625 Communication Systems I
2.4. FOURIER SERIES
� A simple rectangular integration approximation to the aboveintegral is
Xk '1
T0
N�1XnD0
x.nT /e�jk2�.nf0/T0=N �T0
N; k D 0;˙1;˙2; : : :
where N is the number of points used to partition the timeinterval Œ0; T0� and T D T0=N is the time step
� Using the fact that 2�f0T0 D 2� , we can write that
Xk '1
N
N�1XnD0
x.nT /e�j 2�kn
N ; k D 0;˙1;˙2; : : :
� Note that the above must be evaluated for each Fourier coeffi-cient of interest
� Also note that the accuracy of the Xk values depends on thevalue of N
– For k small and x.t/ smooth in the sense that the harmon-ics rolloff quickly,N on the order of 100 may be adequate
– For k moderate, say 5–50, N will have to become in-creasingly larger to maintain precision in the numericalintegral
Calculation Using the FFT
� The FFT is a powerful digital signal processing (DSP) func-tion, which is a computationally efficient version of the dis-crete Fourier transform (DFT)
ECE 5625 Communication Systems I 2-31
CONTENTS
� For the purposes of the problem at hand, suffice it to say thatthe FFT is just an efficient algorithm for computing
XŒk� D
N�1XnD0
xŒn�e�j 2�kn=N ; k D 0; 1; 2; : : : ; N � 1
� If we let xŒn� D x.nT /, then it should be clear that
Xk '1
NXŒk�; k D 0; 1; : : : ;
N
2
� To obtain Xk for k < 0 note that
X�k '1
NXŒ�k� D
1
N
N�1XnD0
x.nT /e�j 2�.�k/n
N
D1
N
N�1XnD0
x.nT /e�j 2�.N�k/n
N D XŒN � k�
since e�j 2�Nn=N D e�j 2�n D 1
� In summary
Xk '
(XŒk�=N; 0 � k � N=2
XŒN � k�=N; �N=2 � k < 0
� To use the Python function fft.fft() to obtain the Xk wesimply let
X D fft.fft(x)
where xD fx.t/ W t D 0; T0=N; 2T0=N; : : : ; T0.N � 1/=N g
� Unlike in MATLAB XŒ0� is really found in X[0]
2-32 ECE 5625 Communication Systems I
2.4. FOURIER SERIES
Example 2.10: Finite Rise/Fall-Time PulseTrain
t
x(t)1
1/2
0 tr T
0τ
τ
τ + tr
Pulse width = τRise and fall time = t
r
Pulse train with finite rise and fall time edges
� Shown above is one period of a finite rise and fall time pulsetrain
� We will numerically compute the Fourier series coefficients ofthis signal using the FFT
� The Python function trap_pulse was written to generate oneperiod of the signal using N samples
def trap_pulse(N,tau,tr):"""xp = trap_pulse(N,tau,tr)
Mark Wickert, January 2015"""n = arange(0,N)t = n/Nxp = zeros(len(t))# Assume tr and tf are equalT1 = tau + tr# Create one period of the trapezoidal pulse waveformfor k in n:
if t[k] <= tr:xp[k] = t[k]/tr
elif (t[k] > tr and t[k] <= tau):
ECE 5625 Communication Systems I 2-33
CONTENTS
xp[k] = 1elif (t[k] > tau and t[k] < T1):
xp[k] = -t[k]/tr + 1 + tau/tr;else:
xp[k] = 0return xp, t
� We now plot the double-sided line spectra for � D 1=8 and twovalues of rise-time tr
2-34 ECE 5625 Communication Systems I
2.4. FOURIER SERIES
1/20
1/8
1/τ = 8
Sidelobes smaller than ideal pulse train which has zero rise time
Signal x.t/ and line spectrum for � D 1=8 and tr D 1=20
� The spectral roll-off rate is faster with the trapezoid
� Clock edges are needed in digital electronics, but by slowingthe edge speed down the clock harmonic generation can bereduced
� The second plot is a more extreme example
ECE 5625 Communication Systems I 2-35
CONTENTS
1/10
1/8
1/τ = 8Sidelobes smaller than with t
r = 1/20
case ~10dB more
Signal x.t/ and line spectrum for � D 1=8 and tr D 1=10
2-36 ECE 5625 Communication Systems I
2.4. FOURIER SERIES
2.4.7 Other Fourier Series Properties
� Given x.t/ has Fourier series (FS) coefficients Xn, if
y.t/ D AC Bx.t/
it follows that
Yn D
(AC BX0; n D 0
BXn; n ¤ 0
proof:
Yn D hy.t/e�j 2�.nf0/ti
D Ahe�j 2�.nf0/ti C Bhx.t/e�j 2�.nf0/ti
D A
�1; n D 0
0; n ¤ 0
�C BXn
QED
� Likewise ify.t/ D x.t � t0/
it follows thatYn D Xne
�j 2�.nf0/t0
proof:Yn D hx.t � t0/e
�j 2�.nf0/ti
Let � D t � t0 which implies also that t D �C t0, so
Yn D hx.�/e�j 2�.nf0/.�Ct0/i
D hx.�/e�j 2�.nf0/�ie�j 2�.nf0/t0
D Xne�j 2�.nf0/t0
QED
ECE 5625 Communication Systems I 2-37
CONTENTS
2.5 Fourier Transform
� The Fourier series provides a frequency domain representationof a periodic signal via the Fourier coefficients and line spec-trum
� The next step is to consider the frequency domain representa-tion of aperiodic signals using the Fourier transform
� Ultimately we will be able to include periodic signals withinthe framework of the Fourier transform, using the concept oftransform in the limit
� The text establishes the Fourier transform by considering alimiting case of the expression for the Fourier series coefficientXn as T0!1
� The Fourier transform (FT) and inverse Fourier transfrom (IFT)is defined as
X.f / D
Z1
�1
x.t/e�j 2�f t dt (FT)
x.t/ D
Z1
�1
X.f /ej 2�f t df (IFT)
� Sufficient conditions for the existence of the Fourier transformare
1.R1
�1jx.t/j dt <1
2. Discontinuities in x.t/ be finite
3. An alternate sufficient condition is thatR1
�1jx.t/j2 dt <
1, which implies that x.t/ is an energy signal
2-38 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
2.5.1 Amplitude and Phase Spectra
� FT properties are very similar to those obtained for the Fouriercoefficients of periodic signals
� The FT, X.f / D Ffx.t/g, is a complex function of f
X.f / D jX.f /jej�.f / D jX.f /jej†X.f /
D RefX.f /g C j ImfX.f /g
� The magnitude jX.f /j is referred to as the amplitude spectrum
� The the angle †X.f / is referred to as the phase spectrum
� Note that
RefX.f /g DZ1
�1
x.t/ cos 2�f t dt
ImfX.f /g DZ1
�1
x.t/ sin 2�f t dt
2.5.2 Symmetry Properties
� If x.t/ is real it follows that
X.�f / D
Z1
�1
x.t/e�j 2�.�f /t dt
D
�Z1
�1
x.t/e�j 2�f t dt
��D X�.f /
thus
jX.�f /j D jX.f /j (even in frequency)†X.�f / D �†X.f / (odd in frequency)
ECE 5625 Communication Systems I 2-39
CONTENTS
� Additionally,
1. For x.�t / D x.t/ (even function), ImfX.f /g D 0
2. For x.�t / D �x.t/ (odd function), RefX.f /g D 0
2.5.3 Energy Spectral Density
� From the definition of signal energy,
E D
Z1
�1
jx.t/j2 dt
D
Z1
�1
x�.t/
�Z1
�1
X.f /ej 2�f t df
�dt
D
Z1
�1
X.f /
�Z1
�1
x�.t/ej 2�f t dt
�df
butZ1
�1
x�.t/ej 2�f t dt D
�Z1
�1
x.t/e�j 2�f t dt
��D X�.f /
� Finally,
E D
Z1
�1
jx.t/j2 dt D
Z1
�1
jX.f /j2 df
which is known as Rayleigh’s Energy Theorem
� Are the units consistent?
– Suppose x.t/ has units of volts
– jX.f /j2 has units of (volts-sec)2
2-40 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
– In a 1 ohm system jX.f /j2 has units of Watts-sec/Hz =Joules/Hz
� The energy spectral density is defined as
G.f /�D jX.f /j2 Joules/Hz
� It then follows that
E D
Z1
�1
G.f / df
Example 2.11: Rectangular Pulse
� Consider
x.t/ D A…
�t � t0
�
�� FT is
X.f / D A
Z t0C�=2
t0��=2
e�j 2�f t dt
D A �e�j 2�f t
�j 2�f
ˇ̌̌̌ˇt0C�=2
t0��=2
D A� �
�ej�f � � e�j�f �
.j 2/�f �
�� e�j 2�f t0
D A�sinc.f �/e�j 2�f t0
A…
�t � t0
�
�F ! A�sinc.f �/e�j 2�f t0
� Plot jX.f /j, †X.f /, and G.f /
ECE 5625 Communication Systems I 2-41
CONTENTS
? 3 ? 2 ? 1 1 2 3
0.2
0.4
0.6
0.8
1
? 3 ? 2 ? 1 1 2 3
? 3
? 2
? 1
1
2
3
? 3 ? 2 ? 1 1 2 3
0.2
0.4
0.6
0.8
1
f
f
f
0-1/τ 1/τ 2/τ-2/τ
0-1/τ 1/τ 2/τ-2/τ
-1/τ 1/τ 2/τ-2/τ
|X(f)|
G(f) = |X(f)|2
X(f)AmplitudeSpectrum
EnergySpectralDensity
PhaseSpectrum
Aτ
(Aτ)2
π
π/2
−π/2
−πt0 = τ/2 slope = -πfτ/2
Rectangular pulse spectra
2.5.4 Transform Theorems
� Be familiar with the FT theorems found in the table of Ap-pendix G.6 of the text
Superposition Theorem
a1x1.t/C a2x2.t/F ! a1X1.f /C a2X2.f /
proof:
2-42 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
Time Delay Theorem
x.t � t0/F ! X.f /e�j 2�f t0
proof:
Frequency Translation Theorem
� In communications systems the frequency translation and mod-ulation theorems are particularly important
x.t/ej 2�f0tF ! X.f � f0/
proof: Note thatZ1
�1
x.t/ej 2�f0te�j 2�f t dt D
Z1
�1
x.t/e�j 2�.f �f0/t dt
soF˚x.t/ej 2�f0t
D X.f � f0/
QED
Modulation Theorem
� The modulation theorem is an extension of the frequency trans-lation theorm
x.t/ cos.2�f0t /F !
1
2X.f � f0/C
1
2X.f C f0/
ECE 5625 Communication Systems I 2-43
CONTENTS
proof: Begin by expanding
cos.2�f0t / D1
2ej 2�f0t C
1
2e�j 2�f0t
Then apply the frequency translation theorem to each term sep-arately
x(t) y(t)
cos(2πf0t)
X(f)Y(f)
f f
A/2
A
f0
-f0
00
signalmultiplier
A simple modulator
Duality Theorem
� Note that
FfX.t/g DZ1
�1
X.t/e�j 2�f t dt D
Z1
�1
X.t/ej 2�.�f /t dt
which implies that
X.t/F ! x.�f /
2-44 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
Example 2.12: Rectangular Spectrum
f-W W0
1X(f)
� Using duality on the above we have
X.t/ D …
�t
2W
�F ! 2W sinc.2Wf / D x.�f /
� Since sinc( ) is an even function (sinc.x/ D sinc.�x/), it fol-lows that
2W sinc.2W t/F ! …
�f
2W
�
Differentiation Theorem
� The general result isd nx.t/
dtnF ! .j 2�f /n X.f /
proof: For n D 1we start with the integration by parts formula,Rudv D uv
ˇ̌̌�Rv du, and apply it to
F�dx
dt
�D
Z1
�1
dx
dte�j 2�f t dt
D x.t/e�j 2�f tˇ̌̌1
�1„ ƒ‚ …0
Cj 2�f
Z1
�1
x.t/e�j 2�f t dt„ ƒ‚ …X.f /
ECE 5625 Communication Systems I 2-45
CONTENTS
alternate — From Leibnitz’s rule for differentiation of inte-grals,
d
dt
Z1
�1
F.f; t/ df D
Z1
�1
@F.f; t/
@fdf
sodx.t/
dtDd
dt
Z1
�1
X.f /ej 2�f t df
D
Z1
�1
X.f /@ej 2�f t
@tdf
D
Z1
�1
j 2�fX.f /ej 2�f t df
) dx=dtF ! j 2�fX.f / QED
Example 2.13: FT of Triangle Pulse
τ−τ 0t
1
� Note that
ττ−τ −τ
t t
1/τ
1/τ
-2/τ-1/τ
2-46 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
� Using the differentiation theorem for n D 2 we have that
F�ƒ
�t
�
��D
1
.j 2�f /2Fn1�ı.t C �/ �
2
�ı.t/C
1
�ı.t � �/
oD
1�ej 2�f � � 2
�C
1�e�j 2�f �
.j 2�f /2
D2 cos.2�f �/ � 2��.2�f /2
D �4 sin2.�f �/4.�f �/2
D �sinc2.f �/
ƒ
�t
�
�F ! �sinc2.f �/
Convolution and Convolution Theorem
� Before discussing the convolution theorem we need to reviewconvolution
� The convolution of two signals x1.t/ and x2.t/ is defined as
x.t/ D x1.t/ � x2.t/ D
Z1
�1
x1.�/x2.t � �/ d�
D x2.t/ � x1.t/ D
Z1
�1
x2.�/x1.t � �/ d�
� A special convolution case is ı.t � t0/
ı.t � t0/ � x.t/ D
Z1
�1
ı.� � t0/x.t � �/ d�
D x.t � �/ˇ̌�Dt0D x.t � t0/
ECE 5625 Communication Systems I 2-47
CONTENTS
Example 2.14: Rectangular Pulse Convolution
� Let x1.t/ D x2.t/ D ….t=�/
� To evaluate the convolution integral we need to consider theintegrand by sketching of x1.�/ and x2.t ��/ on the � axis fordifferent values of t
� For this example four cases are needed for t to cover the entiretime axis t 2 .�1;1/
� Case 1: When t < � we have no overlap so the integrand iszero and x.t/ is zero
λt t + τ/2t - τ/2 0 τ/2−τ/2
x2(t - λ) x
1(λ)
No overlap for t + τ/2 < -τ/2 or t < τ
� Case 2: When �� < t < 0 we have overlap and
x.t/ D
Z1
�1
x1.�/x2.t � �/ d�
D
Z tC�=2
��=2
d� D �ˇ̌̌tC�=2��=2
D t C �=2C �=2 D � C t
Overlap begins when t + τ/2 = -τ/2 or t = -τ
λ
t + τ/20 τ/2−τ/2
x2(t - λ) x
1(λ)
2-48 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
� Case 3: For 0 < t < � the leading edge of x2.t � �/ is to theright of x1.�/, but the trailing edge of the pulse is still over-lapped
x.t/ D
Z �=2
t��=2
d� D �=2 � t C �=2 D � � t
λt + τ/2
t - τ/20 τ/2−τ/2
x2(t - λ)x
1(λ)
Overlap lasts until t = τ
� Case 4: For t > � we have no overlap, and like case 1, theresult is
x.t/ D 0
λt + τ/2t - τ/20 τ/2−τ/2
x2(t - λ)x
1(λ)
No overlap for t > τ
� Collecting the results
x.t/ D
8̂̂̂̂<̂ˆ̂̂:0; t < ��
� C t; �� � t < 0
� � t; 0 � t < �
0; t � �
D
(� � jt j; jt j � �
0; otherwise
ECE 5625 Communication Systems I 2-49
CONTENTS
� Final summary,
…
�t
�
��…
�t
�
�D �ƒ
�t
�
�
� Convolution Theorem: We now consider x1.t/�x2.t/ in termsof the FTZ
1
�1
x1.�/x2.t � �/ d�
D
Z1
�1
x1.�/
�Z1
�1
X2.f /ej 2�f .t��/ df
�d�
D
Z1
�1
X2.f /
�Z1
�1
x1.�/e�j 2�f � d�
�ej 2�f t df
D
Z1
�1
X1.f /X2.f /ej 2�f t df
which implies that
x1.t/ � x2.t/F ! X1.f /X2.f /
Example 2.15: Revisit ….t=�/ �….t=�/
� Knowing that….t=�/�….t=�/ D �ƒ.t=�/ in the time domain,we can follow-up in the frequency domain by writing
F˚….t=�/
� F˚….t=�/
D��sinc.f �/
�2� We have also established the transform pair
�ƒ
�t
�
�F ! �2sinc2.f �/ D �
��sinc2.f �/
�2-50 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
or
ƒ
�t
�
�F ! �sinc2.f �/
Multiplication Theorem
� Having already established the convolution theorem, it followsfrom the duality theorem or direct evaluation, that
x1.t/ � x2.t/F ! X1.f / �X2.f /
2.5.5 Fourier Transforms in the Limit
� Thus far we have considered two classes of signals
1. Periodic power signals which are described by line spec-tra
2. Non-periodic (aperiodic) energy signals which are describedby continuous spectra via the FT
� We would like to have a unifying approach to spectral analysis
� To do so we must allow impulses in the frequency domain byusing limiting operations on conventional FT pairs, known asFourier transforms-in-the-limit
– Note: The corresponding time functions have infinite en-ergy, which implies that the concept of energy spectraldensity will not apply for these signals (we will introducethe concept of power spectral density for these signals)
ECE 5625 Communication Systems I 2-51
CONTENTS
Example 2.16: A Constant Signal
� Let x.t/ D A for �1 < t <1
� We can writex.t/ D lim
T!1A….t=T /
� Note thatF˚A….t=T /
D AT sinc.f T /
� Using the transform-in-the-limit approach we write
Ffx.t/g D limT!1
AT sinc.f T /
? 3 ? 2 ? 1 1 2 3? 0.2
0.2
0.4
0.6
0.8
1
? 3 ? 2 ? 1 1 2 3? 0.2
0.2
0.4
0.6
0.8
1AT1
AT2
T2 >> T
1
f f
Increasing T in AT sinc.f T /
� Note that since x.t/ has no time variation it seems reasonablethat the spectral content ought to be confined to f D 0
� Also note that it can be shown thatZ1
�1
AT sinc.f T / df D A; 8 T
� Thus we have established that
AF ! Aı.f /
2-52 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
� As a further check
F�1˚Aı.f /
D
Z1
�1
Aı.f /ej 2�ff t df D Aej 2�f tˇ̌̌fD0D A
� As a result of the above example, we can obtain several moreFT-in-the-limit pairs
Aej 2�f0tF ! Aı.f � f0/
A cos.2�f0t C �/F !
A
2
�ej�ı.f � f0/C e
�j�ı.f C f0/�
Aı.t � t0/F ! Ae�j 2�f t0
� Reciprocal Spreading Property: Compare
Aı.t/F ! A and A
F ! Aı.f /
A constant signal of infinite duration has zero spectral width,while an impulse in time has zero duration and infinite spectralwidth
2.5.6 Fourier Transforms of Periodic Signals
� For an arbitrary periodic signal with Fourier series
x.t/ D
1XnD�1
Xnej 2�nf0t
ECE 5625 Communication Systems I 2-53
CONTENTS
we can write
X.f / D F"1X
nD�1
Xnej 2�nf0t
#
D
1XnD�1
XnFnej 2�nf0t
oD
1XnD�1
Xnı.f � nf0/
using superposition and FfAej 2�f0tg D Aı.f � f0/
� What is the difference between line spectra and continuousspectra? none!
� Mathematically,
LineSpectra
Convert to time domain
Convert to time domain
Sum phasors
Integrate impulses to get phasors via the inverse FT
ContinuousSpectra
� The Fourier series coefficients need to be known before the FTspectra can be obtained
� A technique that obtains the FT directly will be discussed shortly(page 2–58))
2-54 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
Example 2.17: Ideal Sampling Waveform
� When we discuss sampling theory it will be useful to have theFT of the periodic impulse train signal
ys.t/ D
1XmD�1
ı.t �mTs/
where Ts is the sample spacing or period
� Since this signal is periodic, it must have a Fourier series rep-resentation too
� In particular
Yn D1
Ts
ZTs
ı.t/e�j 2�.nfs/t dt D1
TsD fs; any n
where fs is the sampling rate in Hz
� The FT of ys.t/ is given by
Ys.f / D fs
1XnD�1
F˚ej 2�nf0/t
D fs
1XnD�1
ı.f � nfs/
� Summary,
1XmD�1
ı.t �mTs/F ! fs
1XnD�1
ı.f � nfs/
ECE 5625 Communication Systems I 2-55
CONTENTS
. . . . . .
t
f
ys(t)
Ys(f)
0
0
Ts
fs 4f
s
4Ts
-Ts
-fs
. . . . . .fs
1
An impulse train in time is an impulse train in frequency
Example 2.18: Convolve Step and Exponential
� Find y.t/ D Au.t/ � e�˛tu.t/, ˛ > 0
� For t � 0 there is no overlap so Y.t/ D 0
λ0t
No overlap
2-56 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
� For t > 0 there is always overlap
y.t/ D
Z t
0
A � e�˛.t��/ d�
D Ae�˛t �e˛�
˛
ˇ̌̌t0
D Ae�˛t �e˛t � 1
˛
λ0 t
For t > 0 there is always overlap
� Summary,
y.t/ DA
˛
�1 � e�˛t
�u.t/
t
y(t)
A/α
0
ECE 5625 Communication Systems I 2-57
CONTENTS
Direct Approach for the FT of a Periodic Signal
� The FT of a periodic signal can be found directly by expandingx.t/ as follows
x.t/ D
"1X
mD�1
ı.t �mTs/
#� p.t/ D
1XmD�1
p.t �mTs/
where p.t/ represents one period of x.t/, having period Ts
� From the convolution theorem
X.f / D F(1X
mD�1
ı.t �mTs/
)� P.f /
D fsP.f /
1XnD�1
ı.f � nfs/
D fs
1XnD�1
P.nfs/ı.f � nfs/
where P.f / D Ffp.t/g
� The FT transform pair just established is
1XmD�1
p.t �mTs/F !
1XnD�1
fsP.nfs/ı.f � nfs/
2-58 ECE 5625 Communication Systems I
2.5. FOURIER TRANSFORM
Example 2.19: p.t/ D ….t=2/C….t=4/, T0 D 10
tT
0 = 100 1-1-2 2
. . .. . .
x(t)
1
2
Stacked pulses periodic signal
� We begin by finding P.f / using Ff….t=�/g D �sinc.f �/
P.f / D 2sinc.2f /C 4sinc.4f /
� Plugging into the FT pair derived above with nfs D n=10,
X.f / D1
10
1XnD�1
�2sinc
�n5
�C 4sinc
�2n
5
��ı�f �
n
10
�
2.5.7 Poisson Sum Formula
� The Poisson sum formula from mathematics can be derivedusing the FT pair
e�j 2�.nfs/tF ! ı.f � nfs/
by writing
F�1(1X
nD�1
fsP.nfs/ı.f � nfs/
)D fs
1XnD�1
P.nfs/ej 2�.nfs/t
ECE 5625 Communication Systems I 2-59
CONTENTS
� From the earlier developed FT of periodic signals pair, weknow that the left side of the above is also equal to
1XmD�1
p.t �mTs/alsoD fs
1XnD�1
P.nfs/ej 2�.nfs/t
� We can finally relate this back to the Fourier series coefficients,i.e.,
Xn D fsP.nfs/
2.6 Power Spectral Density and Corre-lation
� For energy signals we have the energy spectral density, G.f /,defined such that
E D
Z1
�1
G.f / df
� For power signals we can define the power spectral density(PSD), S.f / of x.t/ such that
P D
Z1
�1
S.f / df D hjx.t/j2i
– Note: S.f / is real, even and nonnegative
– If x.t/ is periodic S.f / will consist of impulses at theharmonic locations
� For x.t/ D A cos.!0t C �/, intuitively,
S.f / D1
4A2ı.f � f0/C
1
4A2ı.f C f0/
2-60 ECE 5625 Communication Systems I
2.6. POWER SPECTRAL DENSITY AND CORRELATION
sinceRS.f / df D A2=2 as expected (power on a per ohm
basis)
� To derive a general formula for the PSD we first need to con-sider the autocorrelation function
2.6.1 The Time Average Autocorrelation Func-tion
� Let �.�/ be the autocorrelation function of an energy signal
�.�/ D F�1˚G.f /
D F�1
˚X.f /X�.f /
D F�1
˚X.f /
� F�1
˚X�.f /
but x.�t /
F ! X�.f / for x.t/ real, so
�.�/ D x.t/ � x.�t / D
Z1
�1
x.t/x.t C �/ d�
or
�.�/ D limT!1
Z T
�T
x.t/x.t C �/ d�
� Observe thatG.f / D F
˚�.�/
� The autocorrelation function (ACF) gives a measure of the
similarity of a signal at time t to that at time t C � ; the co-herence between the signal and the delayed signal
ECE 5625 Communication Systems I 2-61
CONTENTS
x(t)
X(f)G(f) = |X(f)|2
φ(τ) =
Energy spectral density and signal relationships
2.6.2 Power Signal Case
For power signals we define the autocorrelation function as
Rx.�/ D hx.t/x.t C �/i
D limT!1
1
2T
Z T
�T
x.t/x.t C �/ dt
if periodicD
1
T0
ZT0
x.t/x.t C �/ dt
� Note that
Rx.0/ D hjx.t/j2i D
Z1
�1
Sx.f / df
and since for energy signals �.�/F ! G.f /, a reasonable
assumption is that
Rx.�/F ! Sx.f /
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2.6. POWER SPECTRAL DENSITY AND CORRELATION
� A formal statement of this is the Wiener-Kinchine theorem (aproof is given in text Chapter 7)
Sx.f / D
Z1
�1
Rx.�/e�j 2�f � d�
x(t) Rx(τ) S
x(f)
Power spectral density (PSD) and signal relationships
2.6.3 Properties of R.�/
� The following properties hold for the autocorrelation function
1. R.0/ D hjx.t/j2i � jR.�/j for all values of �
2. R.��/ D hx.t/x.t � �/i D R.�/ ) an even function
3. limj� j!1R.�/ D hx.t/i2 if x.t/ is not periodic
4. If x.t/ is periodic, with period T0, thenR.�/ D R.�CT0/
5. FfR.�/g D S.f / � 0 for all values of f
� The power spectrum and autocorrelation function are frequentlyused for systems analysis with random signals
� For the case of random signal, x.t/, the Fourier transformX.f / is also a random signal, but now in the frequency do-main
� The autocorrelation and PSD of x.t/, under typical assump-tions, are both deterministic functions; this turns out to be vitalin problem solving in Comm II and beyond
ECE 5625 Communication Systems I 2-63
CONTENTS
Example 2.20: Single Sinusoid
� Consider the signal x.t/ D A cos.2�f0t C �/, for all t
Rx.�/ D1
T0
Z T0
0
A2 cos.2�f0t C �/ cos.2�.t C �/C �/ dt
DA2
2T0
ZT0
�cos.2�f0�/C cos.2�.2f0/t C 2�f0� C 2�/
�dt
DA2
2cos.2�f0�/
� Note that
F˚Rx.�/
D Sx.f / D
A2
4
�ı.f � f0/C ı.f C f0/
�
More Autocorrelation Function Properties
� Suppose that x.t/ has autocorrelation function Rx.�/
� Let y.t/ D AC x.t/, A D constant
Ry.�/ D hŒAC x.t/�ŒAC x.t C �/�i
D hA2i C hAx.t C �/i C hAx.t/i C hx.t/x.t C �/i
D A2 C 2Ahx.t/i„ ƒ‚ …const. terms
CRx.�/
� Let z.t/ D x.t � t0/
Rz.�/ D hz.t/z.t C �/i D hx.t � t0/x.t � t0 C �i
D hx.�/x.�C �/i; with � D t � t0D Rx.�/
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2.6. POWER SPECTRAL DENSITY AND CORRELATION
� The last result shows us that the autocorrelation function isblind to time offsets
Example 2.21: Sum of Two Sinusoids
� Consider the sum of two sinusoids
y.t/ D x1.t/C x2.t/
where x1.t/ D A1 cos.2�f1tC�1/ and x2.t/ D A2 cos.2�f2tC�2/ and we assume that f1 ¤ f2
� Using the definition
Ry.�/ D hŒx1.t/C x2.t/�Œx1.t C �/x2.t C �/�i
D hx1.t/x1.t C �/i C hx2.t/x2.t C �/i
C hx1.t/x2.t C �/i C hx2.t/x1.t C �/i
� The last two terms are zero since hcos..!1˙!2/t/i D 0 whenf1 ¤ f2 (why?), hence
Ry.�/ D Rx1.�/CRx2.�/; for f1 ¤ f2
DA212
cos.2�f1�/CA222
cos.2�f2�/
Example 2.22: PN Sequences
� In the testing and evaluation of digital communication systemsa source of known digital data (i.e., ‘1’s and ‘0’s) is required(see also text Chapter 10 p. 524–527)
ECE 5625 Communication Systems I 2-65
CONTENTS
� A maximal length sequence generator or pseudo-noise (PN)code is often used for this purpose
� Practical implementation of a PN code generator can be ac-complished using an N -stage shift register with appropriateexclusive-or feedback connections
� The sequence length or period of the resulting PN code isM D2N � 1 bits long
CD
1Q
1
CD
2Q
2
CD
3Q
3
M = 23 - 1 = 7
one period = MT
t
x(t)
x(t)
ClockPeriod = T
+A
-A
Three stage PN (m-sequence) generator using logic circuits
� PN sequences have quite a number of properties, one being thatthe time average autocorrelation function is of the form shownbelow
2-66 ECE 5625 Communication Systems I
2.6. POWER SPECTRAL DENSITY AND CORRELATION
τ
Rx(τ)
T-T
MT
MT
. . .. . .
A2
-A2/M
PN sequence autocorrelation function
� The calculation of the power spectral density will be left as ahomework problem (a specific example is text Example 2.20)
– Hint: To find Sx.f / D FfRx.�/g we useXn
p.t � nTs/F ! fs
Xn
P.nfs/ı.f � nfs/
where Ts DMT
– One period of Rx.�/ is a triangle pulse with a level shift
� Suppose the logic levels are switched from˙A to positive lev-els of say v1 to v2
– Using the additional autocorrelation function propertiesthis can be done
– You need to know that a PN sequence contains one more‘1’ than ‘0’
� Python code for generating PN sequences from 2 to 12 stagesplus 16, is found ssd.py:
def PN_gen(N_bits,m=5):"""Maximal length sequence signal generator.
ECE 5625 Communication Systems I 2-67
CONTENTS
Generates a sequence 0/1 bits of N_bit duration. The bits themselves are obtainedfrom an m-sequence of length m. Available m-sequence (PN generators) includem = 2,3,...,12, & 16.
Parameters----------N_bits : the number of bits to generatem : the number of shift registers. 2,3, .., 12, & 16
Returns-------PN : ndarray of the generator output over N_bits
Notes-----The sequence is periodic having period 2**m - 1 (2^m - 1).
Examples-------->>> # A 15 bit period signal nover 50 bits>>> PN = PN_gen(50,4)"""c = m_seq(m)Q = len(c)max_periods = int(np.ceil(N_bits/float(Q)))PN = np.zeros(max_periods*Q)for k in range(max_periods):
PN[k*Q:(k+1)*Q] = cPN = np.resize(PN, (1,N_bits))return PN.flatten()
def m_seq(m):"""Generate an m-sequence ndarray using an all-ones initialization.
Available m-sequence (PN generators) include m = 2,3,...,12, & 16.
Parameters----------m : the number of shift registers. 2,3, .., 12, & 16
Returns-------c : ndarray of one period of the m-sequence
Notes-----The sequence period is 2**m - 1 (2^m - 1).
Examples-------->>> c = m_seq(5)"""# Load shift register with all ones to startsr = np.ones(m)# M-squence length is:Q = 2**m - 1c = np.zeros(Q)
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2.6. POWER SPECTRAL DENSITY AND CORRELATION
if m == 2:taps = np.array([1, 1, 1])
elif m == 3:taps = np.array([1, 0, 1, 1])
elif m == 4:taps = np.array([1, 0, 0, 1, 1])
elif m == 5:taps = np.array([1, 0, 0, 1, 0, 1])
elif m == 6:taps = np.array([1, 0, 0, 0, 0, 1, 1])
elif m == 7:taps = np.array([1, 0, 0, 0, 1, 0, 0, 1])
elif m == 8:taps = np.array([1, 0, 0, 0, 1, 1, 1, 0, 1])
elif m == 9:taps = np.array([1, 0, 0, 0, 0, 1, 0, 0, 0, 1])
elif m == 10:taps = np.array([1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1])
elif m == 11:taps = np.array([1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1])
elif m == 12:taps = np.array([1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1])
elif m == 16:taps = np.array([1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1])
else:print ’Invalid length specified’
for n in range(Q):tap_xor = 0c[n] = sr[-1]for k in range(1,m):
if taps[k] == 1:tap_xor = np.bitwise_xor(tap_xor,np.bitwise_xor(int(sr[-1]),int(sr[m-1-k])))
sr[1:] = sr[:-1]sr[0] = tap_xor
return c
R.�/, S.f /, and Fourier Series
� For a periodic power signal, x.t/, we can write
x.t/ D
1XnD�1
Xnej 2�.nf0/t
� There is an interesting linkage between the Fourier series rep-resentation of a signal, the power spectrum, and then back tothe autocorrelation function
ECE 5625 Communication Systems I 2-69
CONTENTS
� Using the orthogonality properties of the Fourier series expan-sion we can write
R.�/ D
* 1X
nD�1
Xnej 2�.nf0/t
! 1X
mD�1
Xmej 2�.mf0/.tC�/
!�+
D
1XnD�1
1XmD�1
XnX�
m
˝ej 2�.nf0t /e�j 2�.mf0/.tC�/
˛„ ƒ‚ …n¤m terms are zero, why?
D
1XnD�1
jXnj2˝ej 2�.nf0/te�j 2�.nf0/.tC�/
˛D
1XnD�1
jXnj2ej 2�.nf0/�
� The power spectral density can be obtained by Fourier trans-forming both sides of the above
S.f / D
1XnD�1
jXnj2ı.f � nf0/
Example 2.23: PN Sequence Analysis and Simulation
� In this example I consider an N D 4 or M D 15 generatorfrom two points of view:
1. First using the FFT to calculate the Fourier coefficientsXn using one period of the waveform (in discrete-time Iuse 10 samples per bit)
2. Second, I just generate a waveform of 10,000 bits (againat 10 samples per bit) and use standard signal processing
2-70 ECE 5625 Communication Systems I
2.6. POWER SPECTRAL DENSITY AND CORRELATION
tools to estimate the time average autocorrelation functionand the PSD
� Numerical Fourier Series Analysis: Generate one period ofthe waveform using ssd.m_seq(4) and then upsample and in-terpolate to create a waveform of 10 samples per bit
� The waveform amplitude levels are Œ0; 1� so there is a largeDC component visible in the spectrum (why?); eight ones andseven zeros makes the average value X0 D 8=15 D 0:533
� The function ssd.m_seq() returns an array of zeros and onesof length 15, i.e., len(x_PN4) = 15
� To create a waveform I upsample the signal by 10 and thenfilter using a finite impulse response of exactly 10 ones
� To plot the line spectrum I use the ssd.line_spectra() func-tion used earlier
ECE 5625 Communication Systems I 2-71
CONTENTS
15 bit period
DC line 0.533 (-5.46 dB)
f = 1/T = 1 HzLine spacing = 1/(MT) = 1/15 Hz
2sinc ( ) spect. shapex
Fourier series based spectral analysis of PN code
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2.6. POWER SPECTRAL DENSITY AND CORRELATION
1 115M
- -=
1–1
15 15MT T= =. . . . . .
Same spectral shape and line spacing as direct FS analysis; scaling with PSD function is different however
DC different as waveform values are [-1,1]
Using simulation to estimate Ry.�/ and Sy.f / of a PN code
� In generating an estimate of the autocorrelation function I usethe FFT to find the time averaged autocorrelation function inthe frequency domain
� In generating the spectral estimate, the Python function psd()(frommatplotlib) function uses Welch’s method of averaged peri-odograms 1
� Here the PSD estimate uses a 4096 point FFT and assumedsampling rate of 10 Hz; the spectral resolution is 10/4096 =0.002441 Hz
1http://en.wikipedia.org/wiki/Periodogram
ECE 5625 Communication Systems I 2-73
CONTENTS
2.7 Linear Time Invariant (LTI) Systems
x(t) y(t) =
operator
Linear system block diagram
Definition
� Linearity (superposition) holds, that is for input ˛1x1.t/C˛2x2.t/,˛1 and ˛2 constants,
y.t/ D H�˛1x1.t/C ˛2x2.t/
�D ˛1H
�x1.t/
�C ˛2H
�x2.t/
�D ˛1y1.t/C ˛2y2.t/
� A system is time invariant (fixed) if for y.t/ D HŒx.t/�, adelayed input gives a correspondingly delayed output, i.e.,
y.t � t0/ D H�x.t � t0/
�Impulse Response and Superposition Integral
� The impulse response of an LTI system is denoted
h.t/�D H
�ı.t/
�assuming the system is initially at rest
� Suppose we can write x.t/ as
x.t/ D
NXnD1
˛nı.t � tn/
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
� For an LTI system with impulse response h. /
y.t/ D
NXnD1
˛nh.t � tn/
� To develop the superposition integral we write
x.t/ D
Z1
�1
x.�/ı.t � �/ d�
' limN!1
NXnD�N
x.n�t/ı.t � n�t/�t; for �t � 1
t
x(t)
0 ∆t−∆t 2∆t 3∆t 4∆t 5∆t 6∆t
Rectangle area is approximation
. . . . . .
Impulse sequence approximation to x.t/
� If we apply H to both sides and let�t ! 0 such that n�t ! �
we have
y.t/ ' limN!1
NXnD�N
x.n�t/h.t � n�t/�t
D
Z1
�1
x.�/h.t � �/ d� D x.t/ � h.t/
orD
Z1
�1
x.t � �/h.�/ d� D h.t/ � x.t/
ECE 5625 Communication Systems I 2-75
CONTENTS
2.7.1 Stability
� In signals and systems the concept of bounded-input bounded-output (BIBO) stability is introduced
� Satisfying this definition requires that every bounded-input (jx.t/j <1) produces a bounded output (jy.t/j <1)
� For LTI systems a fundamental theorem states that a system isBIBO stable if and only ifZ
1
�1
jh.t/j dt <1
� Further implications of this will be discussed later
2.7.2 Transfer Function
� The frequency domain result corresponding to the convolutionexpression y.t/ D x.t/ � h.t/ is
Y.f / D X.f /H.f /
where H.f / is known as the transfer function or frequencyresponse of the system having impulse response h.t/
� It immediately follows that
h.t/F ! H.f /
and
y.t/ D F�1˚X.f /H.f /
D
Z1
�1
X.f /H.f /ej 2�f t df
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
2.7.3 Causality
� A system is causal if the present output relies only on past andpresent inputs, that is the output does not anticipate the input
� The fact that for LTI systems y.t/ D x.t/ � h.t/ implies thatfor a causal system we must have
h.t/ D 0; t < 0
– Having h.t/ nonzero for t < 0 would incorporate futurevalues of the input to form the present value of the output
� Systems that are causal have limitations on their frequency re-sponse, in particular the Paley–Wiener theorem states that forR1
�1jh.t/j2 dt <1, H.f / for a causal system must satisfyZ
1
�1
j ln jH.f /jj1C f 2
df <1
� In simple terms this means:
1. We cannot have jH.f /j D 0 over a finite band of fre-quencies (isolated points ok)
2. The roll-off rate of jH.f /j cannot be too great, e.g., e�k1jf j
and e�k2jf j2
are not allowed, but polynomial forms suchasp1=.1C .f =fc/2N , N an integer, are acceptable
3. Practical filters such as Butterworth, Chebyshev, and el-liptical filters can come close to ideal requirements
ECE 5625 Communication Systems I 2-77
CONTENTS
2.7.4 Properties of H.f /
� For h.t/ real it follows that
jH.�f /j D jH.f /j and †H.�f / D �†H.f /
why?
� Input/output relationships for spectral densities are
Gy.f / D jY.f /j2D jX.f /H.f /j2 D jH.f /j2Gx.f /
Sy.f / D jH.f /j2Sx.f / proof in text chap. 6
Example 2.24: RC Lowpass Filter
C
R
x(t)
X(f ) Y(f )
y(t)
h(t), H(f)
ic(t)
vc(t)
RC lowpass filter schematic
� To find H.f / we may solve the circuit using AC steady-stateanalysis
Y.j!/
X.j!/D
1j!c
RC 1j!c
D1
1C j!RC
so
H.f / DY.f /
X.f /D
1
1C jf=f3; where f3 D 1=.2�RC/
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
� From the circuit differential equation
x.t/ D ic.t/RC y.t/
but
ic.t/ D cdvc.t/
dtD c
y.t/
dt
thus
RCdy.t/
dtC y.t/ D x.t/
� FT both sides using dx=dtF ! j 2�fX.f /
j 2�fRCY.f /C Y.f / D X.f /
so again
H.f / DY.f /
X.f /D
1
1C jf=f3
D1p
1C .f =f3/2e�j tan�1.f =f3/
� The Laplace transform could also be used here, and perhaps ispreferred; we just need to substitute s ! j! ! j 2�f
ECE 5625 Communication Systems I 2-79
CONTENTS
f
ff3
-f3
π/2
-π/2
1
RC lowpass frequency response
� Find the system response to
x.t/ D A…
�.t � T=2/
T
�� Finding Y.f / is easy since
Y.f / D X.f /H.f / D AT sinc.f T /�
1
1C jf=fs
�e�j�f t
� To find y.t/ we can IFT the above, use Laplace transforms, orconvolve directly
� From the FT tables we known that
h.t/ D1
RCe�t=.RC/ u.t/
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
� In Example 2.18 we showed that
Au.t/ � e�˛tu.t/ DA
˛
�1 � e�˛t
�u.t/
� Note that
A…
�t � T=2
T
�D AŒu.t/ � u.t � T /�
and here ˛ D 1=.RC/, so
y.t/ DA
RCRC
�1 � e�t=.RC/
�u.t/
�A
RCRC
�1 � e�.t�T /=.RC/
�u.t � T /
0.5 1 1.5 2 2.5 3
0.2
0.4
0.6
0.8
1
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
t/T fT
fT fT
y(t) |X(f)|, |H(f)|, |Y(f)|
|X(f)|, |H(f)|, |Y(f)| |X(f)|, |H(f)|, |Y(f)|
RC = 2T
RC = T/2RC = T/10
T/10
RC =
T/5
T/2
T
2T
Pulse time response and frequency spectra with A D 1
ECE 5625 Communication Systems I 2-81
CONTENTS
2.7.5 Input/Output with Spectral Densities
� We know that for an LTI system with frequency responseH.f /and input Fourier transform X.f /, the output Fourier trans-form is given by Y.f / D H.f /X.f /
� It is easy to show that in terms of energy spectral density
Gy.f / D jH.f /j2Gx.f /
where Gx.f / D jX.f /j2 and Gy.f / D jY.f /j2
� For the case of power signals a similar relationship holds withthe power spectral density (proof found in Chapter text 7, i.e.,Comm II)
Sy.f / D jH.f /j2 Sx.f /
2.7.6 Response to Periodic Inputs
� When the input is periodic we can write
x.t/ D
1XnD�1
Xnej 2�.nf0/t
which implies that
X.f / D
1XnD�1
Xnı.f � nf0/
� It then follows that
Y.f / D
1XnD�1
XnH.nf0/ı.f � nf0/
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
and
y.t/ D
1XnD�1
XnH.nf0/ej 2�.nf0/t
D
1XnD�1
jXnjjH.nf0jej Œ2�.nf0/tC†XnC†H.nf0/�
� This is a steady-state response calculation, since the analysisassumes that the periodic signal was applied to the system att D �1
2.7.7 Distortionless Transmission
� In the time domain a distortionless system is such that for anyinput x.t/,
y.t/ D H0x.t � t0/
where H0 and t0 are constants
� In the frequency domain the implies a frequency response ofthe form
H.f / D H0e�j 2�f t0;
that is the amplitude response is constant and the phase shift islinear with frequency
� Distortion types:
1. Amplitude response is not constant over a frequency band(interval) of interest$ amplitude distortion
2. Phase response is not linear over a frequency band of in-terest$ phase distortion
ECE 5625 Communication Systems I 2-83
CONTENTS
3. The system is non-linear, e.g., y.t/ D k0 C k1x.t/ C
k2x2.t/$ nonlinear distortion
2.7.8 Group and Phase Delay
� The phase distortion of a linear system can be characterizedusing group delay, Tg.f /,
Tg.f / D �1
2�
d�.f /
df
where �.f / is the phase response of an LTI system
� Note that for a distortionless system �.f / D �2�f t0, so
Tg.f / D �1
2�
d
df� 2�f t0 D t0 s;
clearly a constant group delay
� Tg.f / is the delay that two or more frequency componentsundergo in passing through an LTI system
– If say Tg.f1/ ¤ Tg.f2/ and both of these frequencies arein a band of interest, then we know that delay distortionexists
– Having two different frequency components arrive at thesystem output at different times produces signal disper-sion
� An LTI system passing a single frequency component, x.t/ DA cos.2�f1t /, always appears distortionless since at a single
2-84 ECE 5625 Communication Systems I
2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
frequency the output is just
y.t/ D AjH.f1/j cos�2�f1t C �.f1/
�D AjH1.f /j cos
�2�f1
�t ���.f1/
2�f1
��which is equivalent to a delay known as the phase delay
Tp.f / D ��.f /
2�f
� The system output now is
y.t/ D AjH.f1/j cos�2�f1.t � Tp.f1//�
� Note that for a distortionless system
Tp.f / D �1
2�f.�2�f t0/ D t0
Example 2.25: Terminated Lossless Transmission Line
x(t) y(t)
Rs = R
0
RL = R
0R
0, v
p
L
y.t/ D1
2x�t �
L
vp
�Lossless transmission line
ECE 5625 Communication Systems I 2-85
CONTENTS
� We conclude that H0 D 1=2 and t0 D L=vp
� Note that a real transmission line does have losses that intro-duces dispersion on a wideband signal
Example 2.26: A Fictitious System
-20 -10 10 20
0.5
1
1.5
2
-20 -10 10 20
-1.5
-1
-0.5
0.5
1
1.5
-20 -10 10 20
0.0025
0.005
0.0075
0.01
0.0125
0.015
-20 -10 10 20
0.011
0.012
0.013
0.014
0.015
0.016
f (Hz)
f (Hz)
f (Hz)f (Hz)
H(f)|H(f)|
Tg(f) T
p(f)
No distortion on |f | < 10 Hz band
Ampl. Radians
Time Time
Amplitude, phase, group delay, phase delay
� The system in this example is artificial, but the definitions canbe observed just the same
� For signals with spectral content limited to jf j < 10 Hz thereis no distortion, amplitude or phase/group delay
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
� For 10 < jf j < 20 amplitude distortion is not present, butphase distortion is
� For jf j > 15 both amplitude and phase distortion are present
� What about the interval 10 < jf j < 15?
2.7.9 Nonlinear Distortion
� In the time domain a nonlinear system may be written as
y.t/ D
1XnD0
anxn.t/
� Specifically consider
y.t/ D a1x.t/C a2x2.t/
� Letx.t/ D A1 cos.!1t /C A2 cos.!2t /
� Expanding the output we have
y.t/ D a1�A1 cos.!1t /C A2 cos.!2t /
�C a1
�A1 cos.!1t /C A2 cos.!2t /
�2D a1
�A1 cos.!1t /C A2 cos.!2t /
�C
na22
�A21 C A
22
�Ca2
2
�A21 cos.2!1t /C A22 cos.2!2t /
�oC a2A1A2
˚cosŒ.!1 C !2/t �C cosŒ.!1 � !2/t �
– The third line is the desired output
ECE 5625 Communication Systems I 2-87
CONTENTS
– The fourth line is termed harmonic distortion
– The fifth line is termed intermodulation distortion
f
f f
f
A1
A1
A2
a1A
1
a1A
1
a1A
1A
2
a1A
2a
1a
2A
1
a2A
1
2
2
a2A
1
2
2
a2A
2
2
2a
2(A
1 + A
2)
2 2
2
a2A
1
2
2
2f1
f2
f1
f1
2f1
2f2
f1
f1 f
2-f
1
f1+f
2
f2
0
0
0
0
Non-Linear
Non-Linear
Input
Input
Output
Output
One and two tones in y.t/ D a1x.t/C a2x2.t/ device
� In general if y.t/ D a1x.t/C a2x2.t/ the multiplication theo-rem implies that
Y.f / D a1X.f /C a2X.f / �X.f /
� In particular if X.f / D A…�f=.2W /
�Y.f / D a1A…
�f
2W
�C a22WA
2ƒ
�f
2W
�2-88 ECE 5625 Communication Systems I
2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
Y( f ) =
=
+f f
f
a1A 2Wa
2A2
Wa2A2
-W
-W
W
W
-2W
-2W
2W
2W
a1A + Wa
2A2
a1A + 2Wa
2A2
Continuous spectrum in y.t/ D a1x.t/C a2x2.t/ device
2.7.10 Ideal Filters
1. Lowpass of bandwidth B
HLP.f / D H0…
�f
2B
�e�j 2�f t0
B B-B -B
H0
slope =-2πt
0
|HLP
(f)| HLP
(f)
f f
2. Highpass with cutoff B
HHP.f / D H0
�1 �….f=.2B//
�e�j 2�f t0
B B-B -B
H0
slope =-2πt
0
|HHP
(f)|
f f
HHP
(f)
ECE 5625 Communication Systems I 2-89
CONTENTS
3. Bandpass of bandwidth B
HBP.f / D�Hl.f � f0/CHl.f C f0/
�e�j 2�f t0
where Hl.f / D H0….f=B/
B BH
0
slope = -2πt0
|HBP
(f)| HBP
(f)
f f-f
0f0
-f0
f0
� The impulse response of the lowpass filter is
hLP.t/ D F�1˚H0….f=.2B//e
�j 2�f t0
D 2BH0sincŒ2B.t � t0/�
� Ideal filters are not realizable, but simplify calculations andgive useful performance upper bound results
– Note that hLP.t/ ¤ 0 for t < 0, thus the filter is noncausaland unrealizable
� From the modulation theorem it also follows that
hBP.t/ D 2hl.t � t0/ cosŒ2�f0.t � t0/�D 2BH0sincŒB.t � t0/� cosŒ2�f0.t � t0/�
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
t t
hLP
(t) hBP
(t)
t0
t0
2BH0 2BH
0
t0 - 1
2B t0 - 1
2Bt0 + 1
2B t0 + 1
2B
Ideal lowpass and bandpass impulse responses
2.7.11 Realizable Filters
� We can approximate ideal filters with realizable filters such asButterworth, Chebyshev, and Bessel, to name a few
� We will only consider the lowpass case since via frequencytransformations we can obtain the others
Butterworth
� A Butterworth filter has a maximally flat (flat in the sense ofderivatives of the amplitude response at dc being zero) pass-band
� In the s-domain (s D �Cj!) the transfer function of a lowpassdesign is
HBU.s/ D!nc
.s � s1/.s � s2/ � � � .s � sn/
where
sk D !c exp��
�1
2C2k � 1
2n
��; k D 1; 2; : : : ; n
ECE 5625 Communication Systems I 2-91
CONTENTS
� Note that the poles are located on a semi-circle of radius !c D2�fc, where fc is the 3dB cuttoff frequency of the filter
� The amplitude response of a Butterworth filter is simply
jHBU.f /j D1p
1C .f =fc/2n
Butterworth n D 4 lowpass filter
Chebyshev
� A Chebyshev type I filter (ripple in the passband), is is de-signed to maintain the maximum allowable attenuation in thepassband yet have maximum stopband attenuation
� The amplitude response is given by
jHC.f /j D1p
1C �2C 2n .f /
where
Cn.f / D
(cos.n cos�1.f =fc//; 0 � jf j � fc
cosh.n cosh�1.f =fc//; jf j > fc
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
� The poles are located on an ellipse as shown below
Chebyshev n D 4 lowpass filter
Bessel
� A Bessel filter is designed to maintain linear phase in the pass-band at the expense of the amplitude response
HBE.f / DKn
Bn.f /
where Bn.f / is a Bessel polynomial of order n (see text) andKn is chosen so that the filter gain is unity at f D 0
ECE 5625 Communication Systems I 2-93
CONTENTS
Amplitude Rolloff and Group Delay Comparision
� Compare Butterworth, 0.1 dB ripple Chebyshev, and Bessel
n D 3 Amplitude response
n D 3 Group delay
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
� Despite DSP being pervasive in communication systems, thereis still a great need for analog filter design and implementationtechnologies
� The table below describes some of the well known constructiontechniques
Filter Construction TechniquesConstructionType
Description of El-ements or Filter
Center Fre-quency Range
Unloaded Q
(typicalFilter Appli-cation
LC (passive) lumped elements DC–300 MHzor higher in in-tegrated form
100 Audio, video,IF and RF
Active R, C , op-amps DC–500 kHzor higher usingWB op-amps
200 Audio and lowRF
Crystal quartz crystal 1kHz – 100MHz
100,000 IF
Ceramic ceramic disks withelectrodes
10kHz – 10.7MHz
1,000 IF
Surface acousticwaves (SAW)
interdigitated fin-gers on a Piezo-electric substrate
10-800 MHz, variable IF and RF
Transmission line quarterwave stubs,open and short ckt
UHF and mi-crowave
1,000 RF
Cavity machined andplated metal
Microwave 10,000 RF
Example 2.27: Use Python to Characterize Standard Filters
� You can use the filter design capability of Python with Scipyor MATLAB with the signal processing to study lowpass, band-pass, bandstop, and highpass filters
� Here I will consider two functions written in Python, one fordigital filters and one for analog filters, to allow plotting of gainin dB, phase in radians, and group delay in samples or seconds
ECE 5625 Communication Systems I 2-95
CONTENTS
def freqz_resp(b,a=[1],mode = ’dB’,fs=1.0,Npts = 1024,fsize=(6,4)):"""A method for displaying digital filter frequency response magnitude,phase, and group delay. A plot is produced using matplotlib
freq_resp(self,mode = ’dB’,Npts = 1024)
A method for displaying the filter frequency response magnitude,phase, and group delay. A plot is produced using matplotlib
freqs_resp(b,a=[1],Dmin=1,Dmax=5,mode = ’dB’,Npts = 1024,fsize=(6,4))
b = ndarray of numerator coefficientsa = ndarray of denominator coefficents
Dmin = start frequency as 10**DminDmax = stop frequency as 10**Dmaxmode = display mode: ’dB’ magnitude, ’phase’ in radians, or
’groupdelay_s’ in samples and ’groupdelay_t’ in sec,all versus frequency in Hz
Npts = number of points to plot; defult is 1024fsize = figure size; defult is (6,4) inches
Mark Wickert, January 2015"""f = np.arange(0,Npts)/(2.0*Npts)w,H = signal.freqz(b,a,2*np.pi*f)plt.figure(figsize=fsize)if mode.lower() == ’db’:
plt.plot(f*fs,20*np.log10(np.abs(H)))plt.xlabel(’Frequency (Hz)’)plt.ylabel(’Gain (dB)’)plt.title(’Frequency Response - Magnitude’)
elif mode.lower() == ’phase’:plt.plot(f*fs,np.angle(H))plt.xlabel(’Frequency (Hz)’)plt.ylabel(’Phase (rad)’)plt.title(’Frequency Response - Phase’)
elif (mode.lower() == ’groupdelay_s’) or (mode.lower() == ’groupdelay_t’):"""Notes-----
Since this calculation involves finding the derivative of thephase response, care must be taken at phase wrapping points
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
and when the phase jumps by +/-pi, which occurs when theamplitude response changes sign. Since the amplitude responseis zero when the sign changes, the jumps do not alter the groupdelay results."""theta = np.unwrap(np.angle(H))# Since theta for an FIR filter is likely to have many pi phase# jumps too, we unwrap a second time 2*theta and divide by 2theta2 = np.unwrap(2*theta)/2.theta_dif = np.diff(theta2)f_diff = np.diff(f)Tg = -np.diff(theta2)/np.diff(w)max_Tg = np.max(Tg)#print(max_Tg)if mode.lower() == ’groupdelay_t’:
max_Tg /= fsplt.plot(f[:-1]*fs,Tg/fs)plt.ylim([0,1.2*max_Tg])
else:plt.plot(f[:-1]*fs,Tg)plt.ylim([0,1.2*max_Tg])
plt.xlabel(’Frequency (Hz)’)if mode.lower() == ’groupdelay_t’:
plt.ylabel(’Group Delay (s)’)else:
plt.ylabel(’Group Delay (samples)’)plt.title(’Frequency Response - Group Delay’)
else:s1 = ’Error, mode must be "dB", "phase, ’s2 = ’"groupdelay_s", or "groupdelay_t"’print(s1 + s2)
def freqs_resp(b,a=[1],Dmin=1,Dmax=5,mode = ’dB’,Npts = 1024,fsize=(6,4)):"""A method for displaying analog filter frequency response magnitude,phase, and group delay. A plot is produced using matplotlib
freqs_resp(b,a=[1],Dmin=1,Dmax=5,mode=’dB’,Npts=1024,fsize=(6,4))
b = ndarray of numerator coefficientsa = ndarray of denominator coefficents
Dmin = start frequency as 10**DminDmax = stop frequency as 10**Dmaxmode = display mode: ’dB’ magnitude, ’phase’ in radians, or
’groupdelay’, all versus log frequency in HzNpts = number of points to plot; defult is 1024
ECE 5625 Communication Systems I 2-97
CONTENTS
fsize = figure size; defult is (6,4) inches
Mark Wickert, January 2015"""f = np.logspace(Dmin,Dmax,Npts)w,H = signal.freqs(b,a,2*np.pi*f)plt.figure(figsize=fsize)if mode.lower() == ’db’:
plt.semilogx(f,20*np.log10(np.abs(H)))plt.xlabel(’Frequency (Hz)’)plt.ylabel(’Gain (dB)’)plt.title(’Frequency Response - Magnitude’)
elif mode.lower() == ’phase’:plt.semilogx(f,np.angle(H))plt.xlabel(’Frequency (Hz)’)plt.ylabel(’Phase (rad)’)plt.title(’Frequency Response - Phase’)
elif mode.lower() == ’groupdelay’:"""Notes-----
See freqz_resp() for calculation details."""theta = np.unwrap(np.angle(H))# Since theta for an FIR filter is likely to have many pi phase# jumps too, we unwrap a second time 2*theta and divide by 2theta2 = np.unwrap(2*theta)/2.theta_dif = np.diff(theta2)f_diff = np.diff(f)Tg = -np.diff(theta2)/np.diff(w)max_Tg = np.max(Tg)#print(max_Tg)plt.semilogx(f[:-1],Tg)plt.ylim([0,1.2*max_Tg])plt.xlabel(’Frequency (Hz)’)plt.ylabel(’Group Delay (s)’)plt.title(’Frequency Response - Group Delay’)
else:print(’Error, mode must be "dB" or "phase or "groupdelay"’)
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
� Case 1: A 5th-order Chebyshev type 1 digital bandpass filter,having 1 dB ripple and passband of Œ250; 300� Hz relative to asampling rate of fs D 1000 Hz
5th-Order-Cheby1 BPFDigitalfs = 1000 Hz
Highly peaked near the band-edge, even with 0.1 dB ripple
Digtal bandpass with fs D 1000Hz (Chebyshev)
� The Chebyshev in both analog and digital forms still has a largepeak in the group delay, even with a small ripple (here 0.1 dB)
ECE 5625 Communication Systems I 2-99
CONTENTS
� Case 2: A 7th-order Bessel analog bandpass filter, having pass-band of Œ10; 50� MHz
Not 3dB bandwidth7th-Order Bessel BPFAnalog
Group delay relatively �at in passband, compared with Chebyshev
Analog bandpass (Bessel)
� The Bessel filter has a much lower and smoother group delay,but the magnitude response is rather sloppy
� The filter passband is far from being flat and the roll-off isgradual considering the filter order is seven
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
2.7.12 Pulse Resolution, Risetime, and Band-width
Problem: Given a non-bandlimited signal, what is a reasonable esti-mate of the signals transmission bandwidth?We would like to obtain a relationship to the signals time duration
� Step 1: We first consider a time domain relationship by seekinga constant T such that
T x.0/ D
Z1
�1
jx.t/j dt
t
x(0)
|x(t)|
T/2-T/2 0
Make areas equal via T
� Note thatZ1
�1
x.t/ dt D
Z1
�1
x.t/e�j 2�f t dt
ˇ̌̌̌fD0
D X.0/
and Z1
�1
jx.t/j dt �
Z1
�1
x.t/ dt
which implies
T x.0/ � X.0/
� Step 2: Find a constant W such that
ECE 5625 Communication Systems I 2-101
CONTENTS
2WX.0/ D
Z1
�1
jX.f /j df
f
X(0)
|X(f)|
W-W 0
Make areas equal via W
� Note thatZ1
�1
X.f / df D
Z1
�1
X.f /ej 2�f t df
ˇ̌̌̌tD0
D x.0/
and Z1
�1
jX.f /j df �
Z1
�1
X.f / df
which implies that
2WX.0/ � x.0/
� Combining the results of Step 1 and Step 2, we have
2WX.0/ � x.0/ �1
TX.0/
or
2W �1
Tor W �
1
2T
Example 2.28: Rectangle Pulse
� Consider the pulse x.t/ D ….t=T /
� We know that X.f / D T sinc.f T /
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2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
-1 -0.5 0.5 1
0.2
0.4
0.6
0.8
1
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
fTt /T
x(t) |X(f)|/T
f1/(2T)-1/(2T)
Lower bound for W
Pulse width versus Bandwidth, is W � 1=.2T ?
� We see that for the case of the sinc. / function the bandwidth,W , is clearly greater than the simple bound predicts
Risetime
� There is also a relationship between the risetime of a pulse-likesignal and bandwidth
� Definition: The risetime, TR, is the time required for the lead-ing edge of a pulse to go from 10% to 90% of its final value
� Given the impulse response h.t/ for an LTI system, the stepresponse is just
ys.t/ D
Z1
�1
h.�/u.t � �/ d�
D
Z t
�1
h.�/ d�if causalD
Z t
0
h.�/ d�
Example 2.29: Risetime of RC Lowpass
ECE 5625 Communication Systems I 2-103
CONTENTS
� The RC lowpass filter has impulse response
h.t/ D1
RCe�t=.RC/u.t/
� The step response is
ys.t/ D�1 � e�t=.RC/
�u.t/
� The risetime can be obtained by setting ys.t1/ D 0:1 and ys.t2/ D0:9
0:1 D�1 � e�t1=.RC/
�) ln.0:9/ D
�t1
RC
0:9 D�1 � e�t2=.RC/
�) ln.0:1/ D
�t2
RC
� The difference t2 � t1 is the risetime
TR D t2 � t1 D RC ln.0:9=0:1/ ' 2:2RC D0:35
f3
where f3 is the RC lowpass 3dB frequency
Example 2.30: Risetime of Ideal Lowpass
� The risetime of an ideal lowpass filter is of interest since it isused in modeling and also to see what an ideal filter does to astep input
� The impulse response is
h.t/ D F�1�…
�f
2B
��D 2BsincŒ2Bt�
2-104 ECE 5625 Communication Systems I
2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS
� The step response then is
ys D
Z t
�1
2BsincŒ2B�� d�
D1
�
Z 2�Bt
�1
sinuu
du
D1
2C1
�SiŒ2�Bt�
where Si( ) is a special function known as the sine integral
� We can numerically find the risetime to be
TR '0:44
B
1 2 3 4 5
0.2
0.4
0.6
0.8
1
-2 -1 1 2 3
0.2
0.4
0.6
0.8
1
t RC t/B
Step Response of RC Lowpass Step Response of Ideal Lowpass
2.2 0.44
RC and ideal lowpass risetime comparison
ECE 5625 Communication Systems I 2-105
CONTENTS
2.8 Sampling Theory
Integrate with Chapter 3 material.
2.9 The Hilbert Transform
Integrate with Chapter 3 material.
2.10 The Discrete Fourier Transform andFFT
?
2-106 ECE 5625 Communication Systems I
2.10. THE DISCRETE FOURIER TRANSFORM AND FFT
.
ECE 5625 Communication Systems I 2-107