chapter 2 discrete-time signals and systems. §2.1.1 discrete-time signals: time-domain...
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Chapter 2
Discrete-Time Discrete-Time Signals and SystemsSignals and Systems
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Signals represented as sequences of numbers, called samples
Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the range - n
x[n] defined only for integer values of n and undefined for noninteger values of n
Discrete-time signal represented by {x[n]}
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Discrete-time signal may also be written as a sequence of numbers inside braces:
{x[n]}={…,-0.2,2.2,1.1,0.2,-3.7,2.9,…}
In the above, x[-1]= -0.2, x[0]=2.2, x[1]=1.1, et
c. The arrow is placed under the sample at tim
e index n = 0
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Graphical representation of a discrete-time signal with real-valued samples is as shown below:
§2.1.1 Discrete-Time Signals:Time-Domain Representation
In some applications, a discrete-time sequence {x[n]} may be generated by periodically sampling a continuous-time signal xa(t) at uniform intervals of time
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Here, n-th sample is given by
x[n]=xa(t) |t=nT=xa(nT), n=…,-2,-1,0,1,… The spacing T between two consecutive sam
ples is called the sampling interval or sampling period
Reciprocal of sampling interval T, denoted as FT , is called the sampling frequency:
TFT
1
TFT
1
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Unit of sampling frequency is cycles per second, or hertz (Hz) , if T is in seconds
Whether or not the sequence {x[n]} has been obtained by sampling, the quantity x[n] is called the n-th sample of the sequence
{x[n]} is a real sequence, if the n-th sample x[n] is real for all values of n
Otherwise, {x[n]} is a complex sequence
§2.1.1 Discrete-Time Signals:Time-Domain Representation
A complex sequence {x[n]} can be written as {x[n]}={xre[n]}+j{xim[n]} where xre and xim are the real and imaginary parts of x[n]
The complex conjugate sequence of {x[n]} is given by {x*[n]}={xre[n]}-j{xim[n]}
Often the braces are ignored to denote a sequence if there is no ambiguity
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Example - {x[n]}={cos0.25n} is a real sequence
{y[n]}={ej0.3n} is a complex sequence We can write {y[n]}={cos0.3n + jsin0.3n} ={cos0.3n} + j{sin0.3n}
where {yre[n]}={cos0.3n}
{yim[n]}={sin0.3n}
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Example –
{w[n]}={cos0.3n}- j{sin0.3n}={e-j0.3n} is the complex conjugate sequence of {y[n]}
That is,
{w[n]}= {y*[n]}
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Two types of discrete-time signals: - Sampled-data signals in which samples are continuous-valued- Digital signals in which samples are discrete-valued
Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Example –
Am
pli
tud
eDigital signal
Am
pli
tud
e
Boxedcar signal
Time,t Time,t
§2.1.1 Discrete-Time Signals:Time-Domain Representation
A discrete-time signal may be a finite-length or an infinite-length sequence
Finite-length (also called finite-duration or finite-extent) sequence is defined only for a finite time interval: N1 n N2
where - < N1 and N2 < with N1 N2
Length or duration of the above finite-length sequence is N= N2 - N1+ 1
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Example – x[n]=n2, -3 n 4 is a finite-length sequence of length 4 -(-3)+1=8
y[n]=cos0.4n is an infinite-length sequence
§2.1.1 Discrete-Time Signals:Time-Domain Representation
A length- N sequence is often referred to as an N-point sequence
The length of a finite-length sequence can be increased by zero-padding, i.e., by appending it with zeros
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Example –
is a finite-length sequence of length 12 obtained by zero-padding x[n] =n2, -3≤n≤4 with 4 zero-valued samples
85,0
43,][
2
n
nnnxe
§2.1.1 Discrete-Time Signals:Time-Domain Representation
A right-sided sequence x[n] has zero-valued samples for n < N1
nN1
A right-sided sequence
If N1 0, a right-sided sequence is called a causal sequence
§2.1.1 Discrete-Time Signals:Time-Domain Representation
A left-sided sequence x[n] has zero-valued samples for n > N2
If N2≤0, a left-sided sequence is called a anti-causal sequence
N 2
n
A left-sided sequence
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Size of a Signal
Given by the norm of the signal
Lp - norm
where p is a positive integer
p
n
p
pnxx
/1
][
§2.1.1 Discrete-Time Signals:Time-Domain Representation
The value of p is typically 1 or 2 or ∞
L2 –norm
2x
is the root-mean-squared (rms) value of {x[n]}
§2.1.1 Discrete-Time Signals:Time-Domain Representation
is the peak absolute value of {x[n]}
is the peak absolute value of {x[n]}, i.e.
1xL1 - norm
xL∞ - norm
maxxx
§2.1.1 Discrete-Time Signals:Time-Domain Representation
Example – Let {y[n]}, 0≤n≤N-1, be an approximation of {x
[n]}, 0≤n≤N-1 An estimate of the relative error is given by the
ratio of the L2 -norm of the difference signal and the L2 -norm of {x[n]}:
p
N
n
N
nrel
nx
nxnyE
/1
1
0
2
1
0
2
][
][][
§2.1.2 Operations on Sequences
A single-input, single-output discrete-time system operates on a sequence, called the input sequence, according some prescribed rules and develops another sequence, called the output sequence, with more desirable properties
x[n] y[n]
Input sequence Output sequence
Discrete-timesystem
§2.1.2 Operations on Sequences
For example, the input may be a signal corrupted with additive noise
Discrete-time system is designed to generate an output by removing the noise component from the input
In most cases, the operation defining a particular discrete-time system is composed of some basic operations
§2.2.1 Basic Operations Product (modulation) operation:
x[n] y[n]
w[n]y[n]=x[n].w[n]-Modulator
An application is in forming a finite-length sequence from an infinite-length sequence by multiplying the latter with a finite-length sequence called an window sequence
Process called windowing
§2.2.1 Basic Operations Addition operation:
Ax[n] y[n] y[n]=A.x[n]–Multiplier
Multiplication operation
y[n]=x[n]+w[n]–Adderx[n] y[n]
w[n]
§2.2.1 Basic Operations Time-shifting operation: y[n]=x[n-N] where N
is an integer If N>0, it is delaying operation
1z y[n]x[n] y[n]=x[n-1]
–Unit delay
y[n]x[n] z y[n]=x[n-1]
–Unit advance
If N<0, it is an advance operation
§2.2.1 Basic Operations
Time-reversal (folding) operation:
y[n]=x[n-1] Branching operation: Used to provide multipl
e copies of a sequence
x[n] x[n]
x[n]
§2.2.1 Basic Operations
Example - Consider the two following sequences of length 5 defined for 0n4 :
{a[n]}={3 4 6 –9 0}
{b[n]}={2 –1 4 5 –3} New sequences generated from the above t
wo sequences by applying the basic operations are as follows:
§2.2.1 Basic Operations
{c[n]}={a[n].b[n]}={6 –4 24 –45 0}
{d[n]}={a[n]+b[n]}={5 3 10 –4 -3}
{e[n]}=(3/2){a[n]}={4.5 6 9 –13.5 0} As pointed out by the above example, opera
tions on two or more sequences can be carried out if all sequences involved are of same length and defined for the same range of the time index n
§2.2.1 Basic Operations
However if the sequences are not of same length, in some situations, this problem can be circumvented by appending zero-valued samples to the sequence(s) of smaller lengths to make all sequences have the same range of the time index
Example - Consider the sequence of length 3 defined for 0n 2: {f[n]}={-2, 1, -3}
§2.2.1 Basic Operations
We cannot add the length-3 sequence to the length-5 sequence {a[n]} defined earlier
We therefore first append {f[n]} with 2 zero-valued samples resulting in a length-5 sequence {fe[n]}={-2 1 –3 0 0}
Then
{g[n]}={a[n]}+{f[n]}={1 5 3 –9 0}
§2.2.1 Basic Operations
Ensemble Averaging A very simple application of the addition ope
ration in improving the quality of measured data corrupted by an additive random noise
In some cases, actual uncorrupted data vector s remains essentially the same from one measurement to next
§2.2.1 Basic Operations
While the additive noise vector is random and not reproducible
Let di denote the noise vector corrupting the i-th measurement of the uncorrupted date vector s:
xi=s+di
§2.2.1 Basic Operations
The average data vector, called the ensemble average, obtained after K measurements is given by
For large values of K, xave is usually reasonable replica of the desired data vector s
k
ii
k
ii
k
iiave d
ksds
kx
kx
111
1)(
11
§2.2.1 Basic Operations Example
§2.2.1 Basic Operations
We cannot add the length-3 sequence {f[n]} to the length-5 sequence {a[n]} defined earlier
We therefore first append {f[n]} with 2 zero-valued samples resulting in a length-5 sequence {fe[n]}={−2 1 -3 0 0}
Then
{g[n]}= {g[n]}+{fe[n]}={1 5 3 -9 0}
§2.2.1 Combinations of Basic Operations
Example -
y[n]=1x[n]+ 2x[n-1]+ 3[n-2]+ 4x[n-3]
§2.2.2 Sampling Rate Alteration
Employed to generate a new sequence y[n] with a sampling rate F’T higher or lower than that of the sampling rate FT of a given sequence x[n]
Sampling rate alteration ratio is
R= F’T / FT If R>1, the process called interpolation If R<1, the process called decimation
§2.2.2 Sampling Rate Alteration
In up-sampling by an integer factor L>1, L−1 equidistant zero-valued samples are inserted by the up-sampler between each two consecutive samples of the input sequence x[n]:
otherwise,0
,2,,0],/[][
LLnLnxnxu
x[n] xu[n]L
§2.2.2 Sampling Rate Alteration
An example of the up-sampling
§2.2.2 Sampling Rate Alteration
In down-sampling by an integer factor M>1, every M-th samples of the input sequence are kept and M-1 in-between samples areremoved:
x[n]=x[nM]
x[n] y[n]M
§2.2.2 Sampling Rate Alteration
An example of the down-sampling operation
Classification of SequencesBased on Symmetry
Conjugate-symmetric sequence:
x[n]=-x*[n] If x[n] is real, then it is an even sequence
Classification of SequencesBased on Symmetry
Conjugate-antisymmetric sequence:
x[n]=-x*[-n] If x[n] is real, then it is an odd sequence
Classification of SequencesBased on Symmetry
It follows from the definition that for a conjugate-symmetric sequence {x[n]}, x[0] must be a real number
Likewise, it follows from the definition that for a conjugate anti-symmetric sequence {y[n]}, y[0] must be an imaginary number
From the above, it also follows that for an odd sequence {w[n]}, w[0]=0
Classification of SequencesBased on Symmetry
Any complex sequence can be expressed as a sum of its conjugate-symmetric part and its conjugate-antisymmetric part:
x[n]=xcs[n]+ xca[n]where
xcs[n]=1/2(x[n]+x*[-n])
xca[n]=1/2(x[n]-x*[-n])
Classification of SequencesBased on Symmetry
Example – Consider the length-7 sequence defined for -3≤n≤3
}3,2,65,24,32,41,0{]}[{ jjjjjng
Its conjugate sequence is then given
The time-reversed version of the above
}3,2,65,24,32,41,0{]}[*{ jjjjjng
}0,41,32,24,65,2,3{]}[*{ jjjjjng
Classification of SequencesBased on Symmetry
Likewise
}5.1,5.0,5.15.1,2,5.15.1,5.0,5.1{
]}[*][{2
1]}[{
jjjjj
ngngngca
Therefore
}5.1,35.0,5.45.3,4,5.45.3,35.0,5.1{
]}[*][{2
1]}[{
jjjj
ngngngcs
It can be easily verified that
and ][][ * ngng caca ][][ * ngng cscs
Classification of SequencesBased on Symmetry
Any real sequence can be expressed as a sum of its even part and its odd part:
x[n]=xev[n]+ xod[n]
where
xev[n]=1/2(x[n]+x[-n])
xod[n]=1/2(x[n]-x[-n])
Classification of SequencesBased on Symmetry
A length-N sequence x[n], 0≤n≤N-1, can be expressed as x[n]=xpcs[n]+ xpca[n]
where
xpcs[n]=1/2(x[n]+x*[<-n>N]), 0≤n≤N-1,
is the periodic conjugate-symmetric part and
xpca[n]=1/2(x[n]-x*[<-n>N]), 0≤n≤N-1,
is the periodic conjugate-antisymmetric part
Classification of SequencesBased on Symmetry
For a real sequence, the periodic conjugate- symmetric part, is a real sequence and is called the periodic even part xpe[n]
For a real sequence, the periodic conjugate- antisymmetric part, is a real sequence and is called the periodic odd part xpo[n]
Classification of SequencesBased on Symmetry
A length-N sequence x[n] is called a periodic conjugate-symmetric sequence, if
x[n]=x*[<-n>N ]=x*[<N-n>N ] and is called a periodic conjugate-antisymmetric sequence if
x [n]=-x*[<-n>N])=- x*[<N-n>N ]
Classification of SequencesBased on Symmetry
A finite-length real periodic conjugate- symmetric sequence is called a symmetric sequence
A finite-length real periodic conjugate- antisymmetric sequence is called a antisymmetric sequence
Classification of SequencesBased on Symmetry
Example - Consider the length-4 sequence defined for 0≤n≤3: {u[n]}={1+j4, -2+j3, 4-j2, -5-j6}
Its conjugate sequence is given by {u*[n]}={1-j4, -2-j3, 4+j2, -5+j6}
To determine the modulo-4 time-reversed version {u*[<-n>4]} observe the following:
Classification of SequencesBased on Symmetry
u*[<-0>4]=u*[0]=1-j4
u*[<-1>4]=u*[3]=5+j6
u*[<-2>4]=u*[2]=4+j2
u*[<-3>4]=u*[1]=-2-j3 Hence
{u*[<-n>4]}={1-j4, -5+j6, 4+j2, -2-j3}
Classification of SequencesBased on Symmetry
Therefore
{upcs[n]}=1/2(u[n]+u*[<-n>4])
={1, -3.5+j4.5, 4, -3.5-j4.5} Likewise
{upca[n]}=1/2(u[n]-u*[<-n>4])
={j4, 1.5-j1.5, -2, -1.5-j1.5}
A sequence satisfying][~ nx ][~][~ kNnxnx
Classification of SequencesBased on Symmetry
Smallest value of N satisfying is called the fundamental period
][~][~ kNnxnx
is called a periodic sequence with a period N where N is a positive integer and k is any integer
§2.2.3 Classification of Sequences based on periodicity
Example –
A sequence satisfying the periodicity condition is called an periodic sequence
§2.2.4 Classification of Sequences Energy and Power Signals
Total energy of a sequence x[n] is defined by
nnx 2
x ][ An infinite length sequence with finite sampl
e values may or may not have finite energy A finite length sequence with finite sample v
alues has finite energy
§2.2.4 Classification of Sequences Energy and Power Signals
The average power of an aperiodic sequence is defined by
K
KnKK
nxP 2
121
x ][lim
K
KnKx nx 2
, ][
Define the energy of a sequence x[n] over a finite interval -K n K as
§2.2.4 Classification of Sequences Energy and Power Signals
ThenKx
K KP ,x 12
1lim
The average power of an infinite-length sequence may be finite
1
0
2][~1 N
nx nx
NP
The average power of a periodic sequence
with a period N is given by][~ nx
§2.2.4 Classification of Sequences Energy and Power Signals
Example –Consider the causal sequence defined by
5.412
)1(9lim19
12
1lim
0x
K
K
KP
K
K
nK
Note: x[n] has infinite energy Its average power is given by
0,0
0,)1(3][
n
nnx
n
§2.2.4 Classification of Sequences Energy and Power Signals
An infinite energy signal with finite average power is called a power signal
Example - A periodic sequence which has a finite average power but infinite energy
A finite energy signal with zero average power is called an energy signal
Example - A finite-length sequence which has finite energy but zero average power
Other Types of Classiffications
A sequence x[n] is said to be bounded if
xBnx ][
13.0cos][ nnx
Example - The sequence x[n]=cos(0.3n) is a bounded sequence as
Other Types of Classiffications A sequence x[n] is said to be absolutely sum
mable if
nnx ][
00030
nnny
n
,,.][
is an absolutely summable sequence as
428571
3011
300
..
.n
n
Example - The sequence
Other Types of Classiffications
A sequence x[n] is said to be square-summable if
nnx 2][
nnnh 4.0sin][
is square-summable but not absolutely summable
Example - The sequence
§2.3 Basic Sequences
Unit sample sequence -
0,0
0,1][
n
nn
1
–4 –3 –2 –1 0 1 2 3 4 5 6n
0,0
0,1][
n
nn
–4 –3 –2 –1 0 1 2 3 4 5 6
1
n
Unit step sequence -
§2.3 Basic Sequences
Real sinusoidal sequence -
x[n]=Acos(0n+)
where A is the amplitude, 0 is the angular frequency, and is the phase of x[n]
Example -
§2.3 Basic Sequences Exponential sequence -
,][ nAnx n
,)( oo je , jeAA
],[][][ )( nxjnxeeAnx imrenjj oo
),cos(][ neAnx on
reo
)sin(][ neAnx on
imo
where
then we can express
where A and are real or complex numbers
If we write
§2.3 Basic Sequences xre[n] and xim[n] of a complex exponential seque
nce are real sinusoidal sequences with constant (0=0), growing (0>0) , and decaying (0<0) amplitudes for n > 0
njnx )exp(][612
1
Real part Imaginary part
§2.3 Basic Sequences
Real exponential sequence -
x[n]=An, -< n < where A and are real numbers
=1.2 =0.9
§2.3 Basic Sequences Sinusoidal sequence Acos(0n + ) and comp
lex exponential sequence Bexp(j0n) are periodic sequences of period N if 0N=2rwhere N and r are positive integers
Smallest value of N satisfying 0N=2ris the fundamental period of the sequence
To verify the above fact, consider
x1[n]= Acos(0n + )
x2[n]= Acos(0 ( n+N) + )
§2.3 Basic Sequences
Now
x2[n]= cos(0 ( n+N) + )
= cos(0n+)cos0N - sin(0n+)sin0N
which will be equal to cos(0n+)=x1[n] only if sin0N= 0 and cos0N = 1
These two conditions are met if and only if 0N= 2r or 2/0 = N/r
§2.3 Basic Sequences
If 2/0 is a noninteger rational number, then the period will be a multiple of 2/0
Otherwise, the sequence is aperiodic
Example - is an aperiodic sequence
)3sin(][ nnx
§2.3 Basic Sequences
Here 0=0
Hence period N=2r/0=20 for r=0
§2.3 Basic Sequences
Here 0=0.1
Hence N=2r/0=20 for r=1
0 = 0.1
§2.3 Basic Sequences
Property 1 - Consider x[n]=exp(j1n) and y[n]=exp(j2n) with 0≤ 1< and 2k≤ 2
<2(k +1) where k is any positive integer If 2= 1+2k, then x[n]= y[n] Thus, x[n] and y[n] are indistinguishable
§2.3 Basic Sequences
Property 2 - The frequency of oscillation of Acos(0n) increases as 0 increases from 0 to ,and then decreases as 0 increases from
to 2 Thus, frequencies in the neighborhood of =
0 are called low frequencies, whereas, frequencies in the neighborhood of = are called high frequences
§2.3 Basic Sequences
Because of Property 1, a frequency 0 in the neighborhood =2k is indistinguishable from a frequency 0-2k in the neighborhood of ω=0 and a frequency 0 in the neighborhood of =(2k+1) is indistinguishable from a frequency 0- (2k+1) in the neighborhood of ω=π
§2.3 Basic Sequences
Frequencies in the neighborhood of ω=2πk are usually called low frequencies
Frequencies in the neighborhood of ω=π(2k+1) are usually called high frequencies
v1[n]=cos(0.1πn)= cos(0.9πn) is a low frequency signal
v2[n]=cos(0.8πn)= cos(1.2πn) is a high frequency signal
§2.3 Basic Sequences An arbitrary sequence can be represented in
the time-domain as a weighted sum of some basic sequence and its delayed (advanced) versions
]6[75.0]4[
]2[]1[5.1]2[5.0][
nn
nnnnx
§2.4 The Sampling Process Often, a discrete-time sequence x[n] is devel
oped by uniformly sampling a continuous-time signal xa(t) as indicated below
The relation between the two signals is x[n] =xa(t)|t=nT=xa (nT), n=…, -2, -1, 0, 1, 2, …
§2.4 The Sampling Process
Time variable t of xa(t) is related to the time variable n of x[n] only at discrete-time instants tn given by
with FT=1/T denoting the sampling frequency and T= 2πFT denoting the sampling angular frequency
TTn
n
F
nnTt
2
§2.4 The Sampling Process Consider the continuous-time signal
)cos()2cos()( tAtfAtx oo
is the normalized digital angular frequency of x[n]
ToToo /2where
The corresponding discrete-time signal is
)cos(
)2
cos()cos(][
0
nA
nAnTAnxT
oo
§2.4 The Sampling Process
If the unit of sampling period T is in seconds The unit of normalized digital angular
frequency 0 is radians/sample The unit of normalized analog angular
frequency 0 is radians/second The unit of analog frequency f0 is hertz (Hz)
§2.4 The Sampling Process The three continuous-time signals
)6cos()(1 ttg )14cos()(2 ttg )26cos()(3 ttg
)6.0cos(][1 nng )4.1cos(][2 nng )6.2cos(][3 nng
of frequencies 3Hz, 7Hz, and 13Hz, are sampled at a sampling rate of 10Hz, i.e. with T = 0.1 sec. generating the three sequences
§2.4 The Sampling Process Plots of these sequences (shown with circles)
and their parent time functions are shown below:
0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
1
time
Am
plitu
de
Note that each sequence has exactly the same sample value for any given n
§2.4 The Sampling Process
This fact can also be verified by observing that
)6.0cos()6.02(cos)4.1cos(][2 nnnng
)6.0cos()6.02(cos)6.2cos(][3 nnnng
As a result, all three sequences are identical and it is difficult to associate a unique continuous-time function with each of these sequences
§2.4 The Sampling Process
The above phenomenon of a continuous-time signal of higher frequency acquiring the identity of a sinusoidal sequence of lower frequency after sampling is called aliasing
§2.4 The Sampling Process
Since there are an infinite number of continuous-time signals that can lead to the same sequence when sampled periodically, additional conditions need to imposed so that the sequence {x[n]}={xa[nT]} can uniquely represent the parent continuous-time signal xa(t)
In this case, xa(t) can be fully recovered from {x[n]}
§2.4 The Sampling Process Example - Determine the discrete-time sign
al v[n] obtained by uniformly sampling at a sampling rate of 200Hz the continuous- time signal
Note: va(t) is composed of 5 sinusoidal signals of frequencies 30Hz, 150Hz, 170Hz, 250Hz and 330Hz
)660sin(10)500cos(4)340cos(2)300sin()60cos(6)(
ttttttva
§2.4 The Sampling Process
The sampling period is T=1/200=0.005 sec The generated discrete-time signal v[n] is th
us given by
§2.4 The Sampling Process
Note: v[n] is composed of 3 discrete-time sinusoidal signals of normalized angular frequencies: 0.3π, 0.5π, and 0.7π
)7.0sin(10)6435.05.0cos(5)3.0cos(8)7.0sin(10)5.0cos(4
)3.0cos(2)5.0sin(3)3.0cos(6))7.04sin((10))5.02cos((4
))3.02cos((2))5.02sin((3)3.0cos(6)3.3sin(10)5.2cos(4
)7.1cos(2)5.1sin(3)3.0cos(6][
nnnnn
nnnnn
nnnnn
nnnnv
§2.4 The Sampling Process
Note: An identical discrete-time signal is also generated by uniformly sampling at a 200-Hz sampling rate the following continuous-time signals:
)700sin(3)460cos(6)260sin(10)100cos(4)60cos(2)(
)140sin(10)6435.0100cos(5)60cos(8)(
ttttttg
ttttw
a
a
§2.4 The Sampling Process
Recall 0=20/T
Thus if T>20, then the corresponding normalized digital angular frequency 0 of the discrete-time signal obtained by sampling the parent continuous-time sinusoidal signal will be in the range -<<
Conclusion: No aliasing
§2.4 The Sampling Process
On the other hand, if T < 20 , the normalized digital angular frequency will foldover into a lower digital frequency
0=(20/T)2 in the range -<< because of aliasing
Hence, to prevent aliasing, the sampling frequency T should be greater than 2 times the frequency 0 of the sinusoidal signal being sampled
§2.4 The Sampling Process Generalization: Consider an arbitrary contin
uous-time signal xa(t) composed of a weighted sum of a number of sinusoidal signals
xa(t) can be represented uniquely by its sampled version {x[n]} if the sampling frequency T is chosen to be greater than 2 times the highest frequency contained in xa(t)
§2.4 The Sampling Process
The condition to be satisfied by the sampling frequency to prevent aliasing is called the sampling theorem
A formal proof of this theorem will be presented later
§2.5 Discrete-Time Systems A discrete-time system processes a given
input sequence x[n] to generates an output sequence y[n] with more desirable properties
In most applications, the discrete-time system is a single-input, single-output system:
x[n] y[n]
Input sequence Output sequence
Discrete-TimeSystem
Discrete-Time Systems:Examples
2-input, 1-output discrete-time systems - Modulator, adder
1-input, 1-output discrete-time systems - Multiplier, unit delay, unit advance
Discrete-Time Systems:Examples
Accumulator :
nxny
][][
][]1[][][1
nxnynxxn
The output y[n] at time instant n is the sum of the input sample x[n] at time instant n and the previous output y[n-1] at time instant n-1 which is the sum of all previous input sample values from - to n-1
The system accumulatively adds, i.e., it accumulates all input sample values
Discrete-Time Systems:Examples
Accumulator - Input-output relation can also be written in the form
The second form is used for a causal input sequence, in which case y[-1] is called the initial condition
0,][]1[
][][][
0
0
1
nxy
xxny
n
n
Discrete-Time Systems:Examples
Used in smoothing random variations in data
In most applications, the data x[n] is a bounded sequence
M-point moving-average system –
1
0
][1
][M
k
knxM
ny
M-point average y[n] is also a bounded sequence
Discrete-Time Systems:Examples
If there is no bias in the measurements, an improved estimate of the noisy data is obtained by simply increasing M
A direct implementation of the M-point moving average system requires M−1 additions, 1 division, and storage of M−1 past input data samples
A more efficient implementation is developed next
Discrete-Time Systems:Examples
Hence
1
0
1
1
1
0
][][]1[1
][][][1
][][][1
][
M
M
M
MnxnxnxM
MnxnxnxM
MnxMnxnxM
ny
])[][(1
]1[][ MnxnxM
nyny
Discrete-Time Systems:Examples
Computation of the modified M-point moving average system using the recursive equation now requires 2 additions and 1 division
An application: Consider
x[n] = s[n] + d[n],
where s[n] is the signal corrupted by a noise d[n]
Discrete-Time Systems:Examples
s[n]=2[n(0.9)n], d[n]-random signal
Discrete-Time Systems:Examples Exponentially Weighted Running Average Filter
Computation of the running average requires only 2 additions, 1 multiplication and storage of the previous running average
Does not require storage of past input data samples
10,][]1[][ nxnyny
Discrete-Time Systems:Examples
For 0<α<1, the exponentially weighted average filter places more emphasis on current data samples and less emphasis on past data samples as illustrated below
][]1[]2[]3[][]1[])2[]3[(
][]1[]2[][])1[]2[(][
23
2
2
nxnxnxnynxnxnxny
nxnxnynxnxnyny
Discrete-Time Systems:Examples Linear interpolation - Employed to estimate sa
mple values between pairs of adjacent sample values of a discrete-time sequence
Factor-of-4 interpolation
Discrete-Time Systems:Examples
Factor-of-2 interpolator –
])1[]1[(2
1][][ nxnxnxny uuu
Factor-of-3 interpolator –
])1[]2[(3
2
])2[]1[(3
1][][
nxnx
nxnxnxny
uu
uuu
Discrete-Time Systems:Examples
Factor-of-2 interpolator –
Discrete-Time Systems:Examples
Median Filter – The median of a set of (2K+1) numbers is th
e number such that K numbers from the set have values greater than this number an
d the other K numbers have values smaller Median can be determined by rank-ordering
the numbers in the set by their values and choosing the number at the middle
Discrete-Time Systems:Examples
Median Filter – Example: Consider the set of numbers
{2, -3, 10, 5, -1} Rank-order set is given by
{-3, -1, 2, 5, 10} Hence,
Med{2, -3, 10, 5, -1}=2
Discrete-Time Systems:Examples
Median Filter – Implemented by sliding a window of odd len
gth over the input sequence {x[n]} one sample at a time
Output y[n] at instant n is the median value of the samples inside the window centered at n
Discrete-Time Systems:Examples
Median Filter – Finds applications in removing additive rand
om noise, which shows up as sudden large errors in the corrupted signal
Usually used for the smoothing of signals corrupted by impulse noise
Discrete-Time Systems:Examples Median Filtering Example –
§2.5 Discrete-Time Systems: Classification
Linear System Shift-Invariant System Causal System Stable System Passive and Lossless Systems
§2.5.1 Linear Discrete-Time Systems
Definition - If y1[n] is the output due to an input x1[n] and y2[n] is the output due to an input x2[n] then for an input
x[n] =αx1[n] +βx2[n] the output is given by
y[n] =αy1[n] +βy2[n] Above property must hold for any arbitrary cons
tants α and β and for all possible inputs x1[n] and x2[n]
§2.5.1 Linear Discrete-Time Systems
][][][ 21 nxnxnx For an input
the output is
Hence, the above system is linear
nn
xnyxny
][][,][][ 2211 Accumulator –
nn
n
nynyxx
xxny
][][][][
])[][(][
2121
21
§2.5.1 Linear Discrete-Time Systems
The outputs y1[n] and y2[n] for inputs x1[n] and x2[n] are given by
])[][(]1[][ 20
1
xxynyn
The output y[n] for an input αx1[n]+βx2[n] is given by
nn
xynyxyny0
2220
111 ][]1[][][]1[][
§2.5.1 Linear Discrete-Time Systems
]1[]1[]1[ 21 yyy
Now
][][][ 21 nynyny if Thus
)][][(]1[]1[
)][]1[()][]1[(
])[][
02
0121
022
011
21
nn
nn
xxyy
xyxy
nyny
Now
)][][(]1[]1[
)][]1[()][]1[(
])[][
02
0121
022
011
21
nn
nn
xxyy
xyxy
nyny
§2.5.1 Linear Discrete-Time Systems
For the causal accumulator to be linear the condition y[-1]=αy1[-1]+βy2[-1] must hold for all initial conditions y[-1], y1[-1], y2[-1], and all constants α and β
This condition cannot be satisfied unless the accumulator is initially at rest with zero initial condition
For nonzero initial condition, the system is nonlinear
Nonlinear Discrete-Time Systems
The median filter described earlier is a nonlinear discrete-time system
To show this, consider a median filter with a window of length 3
Output of the filter for an input
{x1[n]}={3, 4, 5}, 0≤n≤2is
{y1[n]}={3, 4, 4}, 0≤n≤2
Nonlinear Discrete-Time Systems
Output for an input
{x2[n]}={2, -1, -1}, 0≤n≤2is
{y2[n]}={0, -1, -1}, 0≤n≤2 However, the output for an input
{x[n]}={x1[n]+ x2[n]}is
{y[n]}={3, 4, 3}
Nonlinear Discrete-Time Systems
Note
{y1[n]+y2[n]}={3, 3, 3}≠{y [n]} Hence, the median filter is a nonlinear discre
te-time system
§2.5.1 Shift-Invariant System For a shift-invariant system, if y1[n] is the re
sponse to an input x1[n] , then the response to an input x[n]=x1[n-n0]
is simply y[n]=y1[n-n0]
where n0 is any positive or negative integer The above relation must hold for any arbitrar
y input and its corresponding output The above property is called time-invariance
property, or shift-invariant proterty
§2.5.1 Shift-Invariant System
In the case of sequences and systems with indices n related to discrete instants of time, the above property is called time-invariance property
Time-invariance property ensures that for a specified input, the output is independent of the time the input is being applied
§2.5.1 Shift-Invariant System Example - Consider the up-sampler with an i
nput-output relation given by
otherwise,0
,2,,0,]/[][
LLnLnxnxu
For an input x1[n]=x[n-n0] the output x1,u[n] is given by
otherwise,0
,2,,0,]/)[(otherwise,0
,2,,0,]/[][
0
1,1
LLnLLnnx
LLnLnxnx u
§2.5.1 Shift-Invariant System
However from the definition of the up-sampler
Hence, the up-sampler is a time-varying system
][otherwise,0
,2,,,/][][
,1
0000
0
nx
LnLnnnLnnxnnx
u
u
§2.5.2 Linear Time-Invariant system
Linear Time-Invariant (LTI) System - A system satisfying both the linearity and
the time-invariance property LTI systems are mathematically easy to
analyze and characterize, and consequently, easy to design
Highly useful signal processing algorithms have been developed utilizing this class of systems over the last several decades
Causal System
In a causal system, the n0-th output sampley[n0] depends only on input samples x[n] forn≤n0 and does not depend on input samples for n>n0
Let y1[n] and y2[n] be the responses of acausal discrete-time system to the inputs x1
[n] and x2[n], respectively
Causal System
Then
x1[n]= x2[n] for n<N
implies also that
y1[n]= y2[n] for n<N For a causal system, changes in output sam
ples do not precede changes in the input samples
Causal System Examples of causal systems:
y[n]=α1x[n]+α2x[n-1]+α3x[n-2]+α4x[n-3]
y[n]=b0x[n]+b1x[n-1]+b2x[n-2]
+a1y[n-1]+a2y[n-2]y[n]=y[n-1]+x[n]
Examples of noncausal systems:
y[n]=xu[n]+1/2(xu[n-1]+ xu[n+1])
y[n]=xu[n]+1/3(xu[n-1]+ xu[n+2])
+ 2/3(xu[n-2]+ xu[n+1])
Causal System
A noncausal system can be implemented as a causal system by delaying the output by an appropriate number of samples
For example a causal implementation of the factor-of-2 interpolator is given by
y[n]=xu[n-1]+1/2(xu[n-2]+ xu[n])
Stable System
There are various definitions of stability We consider here the bounded-input, bound
ed-output (BIBO) stability If y[n] is the response to an input x[n] and if
|x[n]|≤Bx for all values of n Then
|y[n]|≤By for all values of n
Stable System
Example - The M-point moving average filter is BIBO stable:
For a bounded input |x[n]|≤Bx we have
1
0
][1
][M
k
knxM
ny
xx
M
k
M
k
BMBM
knxM
knxM
ny
)(1
][1
][1
][1
0
1
0
§2.5.3 Passive and Lossless Systems
A discrete-time system is defined to be passive if, for every finite-energy input x[n], the output y[n] has, at most, the same energy, i.e.
nnnxny 22 ][][
For a lossless system, the above inequality is satisfied with an equal sign for every input
§2.5.3 Passive and Lossless Systems
Example - Consider the discrete-time system defined by y[n]=x[n-N] with N a positive integer
Its output energy is given by
nnnxny 222 ][][
Hence, it is a passive system if || 1 and is a lossless system if || =1
§2.5.4 Impulse and Step Responses
The response of a discrete-time system to a unit sample sequence {δ[n]} is called the unit impulse response or simply, the impulse response, and is denoted by {h[n]}
The response of a discrete-time system to a unit step sequence {μ[n]} is called the unit step response or simply, the step response, and is denoted by {s[n]}
§2.5.4 Impulse and Step Responses
Example - The impulse response of the system
y[n]=a1x[n]+a2x[n-1]+a3x[n-2]+a4x[n-3]
is obtained by setting x[n]=δ[n] resulting in
h[n]=a1δ[n]+a2δ[n-1]+a3δ[n-2]+a4δ[n-3] The impulse response is thus a finite-length
sequence of length 4 given by{h[n]={a1, a2, a3, a4}
§2.5.4 Impulse and Step Responses
Example - The impulse response of the discrete-time accumulator
n
xny
][][
][][][ nnhn
is obtained by setting x[n] = δ[n] resulting in
§2.5.4 Impulse and Step Responses
Example - The impulse response {h[n]} of the factor-of-2 interpolator
])[][(][][ 1121 nxnxnxny uuu
])[][(][][ 1121 nnnnh
}.,.{]}[{ 50150
nh
is obtained by setting xu[n]= [n] and is given by
The impulse response is thus a finite-length sequence of length 3:
§2.6 Time-Domain Characterization of LTI Discrete-Time System
Input-Output Relationship –
A consequence of the linear, time-invariance property is that an LTI discrete-time system is completely characterized by its impulse response
Knowing the impulse response one can compute the output of the system for any arbitrary input
§2.6 Time-Domain Characterization of LTI Discrete-Time System
Let h[n] denote the impulse response of a LTI discrete-time system
Compute its output y[n] for the input:
]5[75.0]2[]1[5.1]2[5.0][ nnnnnx
As the system is linear, we can compute its outputs for each member of the input separately and add the individual outputs to determine y[n]
§2.6 Time-Domain Characterization of LTI Discrete-Time System
Since the system is time-invariant
]5[]5[]2[]2[
]1[]1[]2[]2[
outputinput
nhnnhnnhnnhn
§2.6 Time-Domain Characterization of LTI Discrete-Time System
Likewise, as the system is linear
][.][.][ 151250 nhnhny][.][ 57502 nhnh
Hence because of the linearity property we get Likewise, as the system is linear
]5[75.0]5[75.0]2[]2[]1[5.1]1[5.1]2[5.0]2[5.0
outputinput
nhnnhn
nhnnhn
§2.6 Time-Domain Characterization of LTI Discrete-Time System
Now, any arbitrary input sequence x[n] can be expressed as a linear combination of delayed and advanced unit sample sequences in the form
kknkxnx ][][][
The response of the LTI system to an input x[k][n-k] will be x[k]h[n-k]
§2.6 Time-Domain Characterization of LTI Discrete-Time System
Hence, the response y[n] to an input
kknkxnx ][][][
kknhkxny ][][][
kkhknxny ][][][
which can be alternately written as
will be
§2.6.1 Convolution Sum
The summation
kknhknxknhkxny ][][][][][
y[n] = x[n] h[n]*
is called the convolution sum of the sequences x[n] and h[n] and represented compactly as
§2.6.1 Convolution SumProperties -
Commutative property:
x[n] h[n] = h[n] x[n]* *
(x[n] h[n]) y[n] = x[n] (h[n] y[n])****
x[n] (h[n] + y[n]) = x[n] h[n] + x[n] y[n]** *
Associative property :
Distributive property :
§2.6.1 Convolution Sum
Interpretation - 1) Time-reverse h[k] to form h[-k] 2) Shift h[-k] to the right by n sampling perio
ds if n > 0 or shift to the left by n sampling periods if n < 0 to form h[n-k]
3) Form the product v[k]=x[k]h[n-k] 4) Sum all samples of v[k] to develop the
n-th sample of y[k] of the convolution sum
§2.6.1 Convolution Sum
Schematic Representation -
nz][ knh
][ kh
][kx
][kv][ny
k
The computation of an output sample using the convolution sum is simply a sum of products
Involves fairly simple operations such as additions, multiplications, and delays
§2.6.1 Convolution Sum We illustrate the convolution operation for th
e following two sequences:
Figures on the next several slides the steps involved in the computation of
otherwise,0
50,1][
nnx
otherwise,0
50,3.08.1][
nnnh
y[n]= x[n] h[n]*
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
§2.6.1 Convolution Sum
Time-Domain Characterization of LTI Discrete-Time System
In practice, if either the input or the impulse response is of finite length, the convolution sum can be used to compute the output sample as it involves a finite sum of products
If both the input sequence and the impulse response sequence are of finite length, the output sequence is also of finite length
Time-Domain Characterization of LTI Discrete-Time System
If both the input sequence and the impulse response sequence are of infinite length, convolution sum cannot be used to compute the output
For systems characterized by an infinite impulse response sequence, an alternate time-domain description involving a finite sum of products will be considered
Time-Domain Characterization of LTI Discrete-Time System
Example - Develop the sequence y[n] generated by the convolution of the sequences x[n] and h[n] shown below
Time-Domain Characterization of LTI Discrete-Time System
As can be seen from the shifted time-reversed version {h[n-k]} for n<0, shown below for n =-3 , for any value of the sample index k, the k-th sample of either {x[k]} or is zero
Time-Domain Characterization of LTI Discrete-Time System
As a result, for n<0, the product of the k-th samples of {x[k]} and {h[n-k]} is always zero, and hence y[n] = 0 for n < 0
Consider now the computation of y[0]
The sequence {h[n-k]} is shown on the right
Time-Domain Characterization of LTI Discrete-Time System
The product sequence {x[k]h[-k]} is plotted below which has a single nonzero sample x[0]h[0] for k=0
Thus y[0]=x[0]x[0]=-2
Time-Domain Characterization of LTI Discrete-Time System
For the computation of y[1], we shift {h[-k]} to the right by one sample period to form {h[1-k]} as shown below on the left
The product sequence {x[k]h[1-k]} is shown below on the right
Hence, y[1]=x[0]h[1]+x[1]h[0]=-4+0=-4
Time-Domain Characterization of LTI Discrete-Time System
To calculate y[2], we form as shown below on the left -6
The product sequence {x[k]h[2-k]} is plotted below on the right
y[2]=x[0]h[2]+x[1]h[1]+x[2]h[0]=0+0+1=1
Time-Domain Characterization of LTI Discrete-Time System
Continuing the process we get y[3]=x[0]h[3]+x[1]h[2]+x[2]h[1]+x[3]h[0] =2+0+0+1=3 y[4]=x[1]h[3]+x[2]h[2]+x[3]h[1]+x[4]h[0] =0+0-2+3=1 y[5]=x[2]h[3]+x[3]h[2]+x[4]h[1]=-1+0+6=5 y[6]=x[3]h[3]+x[4]h[2]=1+0=1 y[6]=x[4]h[3]=-3
Time-Domain Characterization of LTI Discrete-Time System
From the plot of {h[n-k]} for n > 7 and theplot of {x[k]} as shown below, it can be seen that there is no overlap between these two sequences
As a result y[n] = 0 for n > 7
Time-Domain Characterization of LTI Discrete-Time System
The sequence {y[n]} generated by the convolution sum is shown below
Time-Domain Characterization of LTI Discrete-Time System
Note: The sum of indices of each sample product inside the convolution sum is equal to the index of the sample being generated by the convolution operation
For example, the computation of y[3] in the previous example involves the products x[0]h[3], x[1]h[2], x[2]h[1], and x[3]h[0]
The sum of indices in each of these products is equal to 3
Time-Domain Characterization of LTI Discrete-Time System
In the example considered the convolution of a sequence {x[n]} of length 5 with a sequence {h[n]} of length 4 resulted in a sequence {y[n]} of length 8
In general, if the lengths of the two sequences being convolved are M and N, then the sequence generated by the convolution is of length M+N-1
Tabular Method ofConvolution Sum Computation
Can be used to convolve two finite-length sequences
Consider the convolution of {g[n]}, 0≤n≤3, with {h[n]}, 0≤n≤2, generating the
Samples of {g[n]} and {h[n]} are then multiplied using the conventional multiplication method without any carry operation
sequence y[n]= g[n] h[n]*
Tabular Method ofConvolution Sum Computation
The samples y[n] generated by the convolution sum are obtained by adding the entries in the column above each sample
]5[]4[]3[]2[]1[]0[:][
]2[]3[]2[]2[]2[]1[]2[]0[]1[]3[]1[]2[]1[]1[]1[]0[
]0[]3[]0[]2[]0[]1[]0[]0[]2[]1[]0[:][
]3[]2[]1[]0[:][543210:
yyyyyyny
hghghghghghghghg
hghghghghhhnh
ggggngn
Tabular Method ofConvolution Sum Computation
The samples of {y[n]} are given by y[0]=g[0]h[0] y[1]=g[1]h[0]+g[0]h[1] y[2]=g[2]h[0]+g[1]h[1]+g[0]h[2] y[3]=g[3]h[0]+g[2]h[1]+g[1]h[2] y[4]=g[3]h[1]+g[2]h[2] y[5]=g[3]h[2]
Tabular Method ofConvolution Sum Computation
The method can also be applied to convolve two finite-length two-sided sequences
In this case, a decimal point is first placed to the right of the sample with the time index n = 0 for each sequence
Next, convolution is computed ignoring the location of the decimal point
Tabular Method ofConvolution Sum Computation
Finally, the decimal point is inserted according to the rules of conventional multiplication
The sample immediately to the left of the decimal point is then located at the time index n = 0
Convolution Using MATLAB
The M-file conv implements the convolution sum of two finite-length sequences
If a=[-2 0 1 -1 3]
b=[1 2 0 -1]
then conv(a,b) yields
[-2 -4 1 3 1 5 1 -3]
Simple InterconnectionSchemes
Two simple interconnection schemes are: Cascade Connection Parallel Connection
Cascade Connection
Impulse response h[n] of the cascade of two LTI discrete-time systems with impulse responses h1[n] and h2[n] is given by
y[n]= h1[n] h2[n]*
][nh1][nh2][nh1 ][nh2
][][ nhnh 1 ][nh2][nh1 *
Cascade Connection
Note: The ordering of the systems in the cascade has no effect on the overall impulse response because of the commutative property of convolution
A cascade connection of two stable systems is stable
A cascade connection of two passive (lossless) systems is passive (lossless)
Cascade Connection
An application is in the development of an inverse system
If the cascade connection satisfies the relation
then the LTI system h1[n] is said to be the inverse of h2[n] and vice-versa
=[n]
h1[n] h2[n]*
If the impulse response of the channel is known, then x[n] can be recovered by designing an inverse system of the channel
Cascade Connection An application of the inverse system concept
is in the recovery of a signal x[n] from its distorted version appearing at the output of a transmission channel
][ˆ nx
=[n]
h1[n] h2[n]*
][nh2][nh1x[n]
x[n]
][ˆ nxchannel Inverse system
Cascade Connection
Example - Consider the discrete-time accumulator with an impulse response µ[n]
Its inverse system satisfy the condition It follows from the above that h2[n]=0 for n<0
and
1for0][
1]0[
02
2
nh
hn
Cascade Connection
Thus the impulse response of the inverse system of the discrete-time accumulator is given by
h2[n]=[n]- [n-1]
which is called a backward difference system
Parallel Connection
Impulse response h[n] of the parallel connection of two LTI h1[n] discrete-time systems with impulse responses and h2[n] is given by
h[n]=h1[n]+h2[n]
][nh2
][nh1 ][][ nhnh 1 ][nh2][nh1
§2.6.2 Simple Interconnection Schemes
h1[n]=[n]+0.5[n-1],
h2[n]=0.5[n]-0.25[n-1],
h3[n]=2[n],
h4[n]=- 2(0.5)n[n]
][nh2
][nh1
][nh4
][nh3
Consider the discrete-time system where
§2.6.2 Simple Interconnection Schemes
Simplifying the block-diagram we obtain
][nh2
][nh1
][][ 43 nhnh
][nh1
])[][(][ 432 nhnhnh *
§2.6.2 Simple Interconnection Schemes
Overall impulse response h[n] is given by
][][][][][ nhnhnhnhnh 42321 ])[][(][][][ nhnhnhnhnh 4321 *
* *
][2])1[][(][][41
21
32 nnnnhnh
]1[][21 nn
* *
Now,
§2.6.2 Simple Interconnection Schemes
Therefore][][]1[
2
1][]1[
2
1][][ nnnnnnnh
][)
2
1(2])1[
4
1][
2
1(][][ 42 nnnnhnh n* *
][][)2
1(
]1[)2
1(][)
2
1(
]1[)2
1(
2
1][)
2
1( 1
nn
nn
nn
n
nn
nn
Stability Condition of an LTI Discrete-Time System
BIBO Stability Condition - A discrete- time is BIBO stable if and only if the output sequence {y[n]} remains bounded for all bounded input sequence {x[n]}
An LTI discrete-time system is BIBO stable if and only if its impulse response sequence {h[n]} is absolutely summable, i.e.
n
nhS ][
Stability Condition of an LTI Discrete-Time System
Proof: Assume h[n] is a real sequence Since the input sequence x[n] is bounded w
e have xBnx ][
Therefore
SBkhB
knxkhknxkhny
xk
x
kk
][
][][][][][
Stability Condition of an LTI Discrete-Time System
Thus, S < ∞ implies |y[n]|≤By < ∞ indicating that y[n] is also bounded
To prove the converse, assume y[n] is bounded, i.e., |y[n]|≤By
Consider the input given by
0][if,
0][if}),{sgn(][
nhK
nhnhnx
Stability Condition of an LTI Discrete-Time System
where sgn(c) = +1 if c > 0 and sgn(c) =-1 if c < 0 and |K| ≤ 1
Note: Since |x[n]| ≤ 1, is obviously bounded For this input, y[n] at n = 0 is
Therefore, |y[n]|≤By implies S < ∞
k
yBSkhkhy ][])[sgn(]0[
Stability Condition of an LTI Discrete-Time System
Example - Consider a causal LTI discrete- time system with an impulse response h[n]=(α)nµ[n]
For this system
Therefore S < ∞ if |α|<1 for which the system is BIBO stable
If |α|<1, the system is not BIBO stable
1if1
1][
0
n
n
n
n nS
Causality Condition of an LTI Discrete-Time System
Let x1[n] and x2[n] be two input sequences with
x1[n]=x2[n] for n≤n0
The corresponding output samples at n=n0 of an LTI system with an impulse response {h[n]} are then given by
Causality Condition of an LTI Discrete-Time System
][][
][][][][][
01
1
010
0101
knxkh
knxkhknxkhny
k
kk
][][
][][][][][
02
1
020
0202
knxkh
knxkhknxkhny
k
kk
Causality Condition of an LTI Discrete-Time System
If the LTI system is also causal, then
y1[n0]=y2[n0]
As x1[n]=x2[n] for n≤n0
][][][][ 020
010
knxkhknxkhkk
This implies
][][][][ 02
1
01
1
knxkhknxkhkk
Causality Condition of an LTI Discrete-Time System
As x1[n] ≠x2[n] for n>n0 the only way the condition
will hold if both sums are equal to zero, which is satisfied if h[k] = 0 for k < 0
][][][][ 02
1
01
1
knxkhknxkhkk
Causality Condition of an LTI Discrete-Time System
]3[]2[]1[][][ 4321 nxnxnxnxny
An LTI discrete-time system is causal if and only if its impulse response {h[n]} is a causal sequence
Example - The discrete-time system defined by
is a causal system as it has a causal impulse response
}{]}[{ 4321 nh
Example - The discrete-time accumulator defined by
Causality Condition of an LTI Discrete-Time System
is a causal system as it has a causal impulse response given by
][][][ nnyn
][][][ nnhn
Causality Condition of an LTI Discrete-Time System
Example - The factor-of-2 interpolator defined by
])1[]1[(2
1][][ nxnxnxny uuu
is noncausal as it has a noncausal impulse response given by
}5.015.0{]}[{ nh
Causality Condition of an LTI Discrete-Time System
Note: A noncausal LTI discrete-time system with a finite-length impulse response can often be realized as a causal system by inserting an appropriate amount of delay
For example, a causal version of the factor- of-2 interpolator is obtained by delaying the input by one sample period:
])[]2[(2
1]1[][ nxnxnxny uuu
Finite-Dimensional LTI Discrete-Time Systems
An important subclass of LTI discrete-time systems is characterized by a linear constant coefficient difference equation of the form
x[n] and y[n] are, respectively, the input and the output of the system
{dk} and {pk} are constants aracterizing the system
M
kk
N
kk knxpknyd
00
][][
Finite-Dimensional LTI Discrete-Time Systems
The order of the system is given by max(N,M), which is the order of the difference equation
It is possible to implement an LTI system characterized by a constant coefficient difference equation as here the computation involves two finite sums of products
Finite-Dimensional LTI Discrete-Time Systems
If we assume the system to be causal, then the output y[n] can be recursively computed using
provided d0≠0
y[n] can be computed for all n≥n0, knowing x[n] and the initial conditions
y[n0-1], y[n0-2],…, y[n0-N]
M
k
kN
k
k knxd
pkny
d
dny
0 01 0
][][][
§2.7 Classification of LTI Discrete-Time Systems
Based on Impulse Response Length - If the impulse response h[n] is of finite length,
i.e.,
h[n]=0 for N1<n<N2 and N1<N2
then it is known as a finite impulse response (FIR) discrete-time system
The convolution sum description here is
2
1
][][][N
Nkknxkhny
§2.7 Classification of LTI Discrete-Time Systems
The output y[n] of an FIR LTI discrete-time system can be computed directly from the convolution sum as it is a finite sum of products
Examples of FIR LTI discrete-time systems are the moving-average system and the linear interpolators
§2.7 Classification of LTI Discrete-Time Systems
If the impulse response is of infinite length, then it is known as an infinite impulse response (IIR) discrete-time system
The class of IIR systems we are concerned with in this course are characterized by linear constant coefficient difference equations
§2.7 Classification of LTI Discrete-Time Systems
Example - The discrete-time accumulator defined by
y[n]=y[n-1]+x[n]
is seen to be an IIR system
§2.7 Classification of LTI Discrete-Time Systems
Example - The familiar numerical integration formulas that are used to numerically solve integrals of the form
t
dxty0
)()(
can be shown to be characterized by linear constant coefficient difference equations, and hence, are examples of IIR systems
§2.7 Classification of LTI Discrete-Time Systems
If we divide the interval of integration into n equal parts of length T, then the previous integral can be rewritten as
nT
Tn
dxTnynTy)1(
)())1(()(
nT
dxnTy0
)()(
where we have set t = nT and used the notation
§2.7 Classification of LTI Discrete-Time Systems
Using the trapezoidal method we can write
)}())1(({)(2
)1(
nTxTnxdx TnT
Tn
)}())1(({))1(()(2
nTxTnxTnynTy T
Hence, a numerical representation of the definite integral is given by
§2.7 Classification of LTI Discrete-Time Systems
Let y[n] = y(nT) and x[n] = x(nT) Then
)}())1(({))1(()(2
nTxTnxTnynTy T
]}1[][{]1[][2
nxnxnyny T
which is recognized as the difference equation representation of a first-order IIR discrete-time system
reduces to
§2.7 Classification of LTI Discrete-Time Systems
Based on the Output Calculation Process Nonrecursive System - Here the output can
be calculated sequentially, knowing only the present and past input samples
Recursive System - Here the output computation involves past output samples in addition to the present and past input samples
§2.7 Classification of LTI Discrete-Time Systems
Based on the Coefficients - Real Discrete-Time System - The impulse re
sponse samples are real valued Complex Discrete-Time System - The impul
se response samples are complex valued
§2.8 Correlation of Signals
There are applications where it is necessary to compare one reference signal with one or more signals to determine the similarity between the pair and to determine additional information based on the similarity
§2.8 Correlation of Signals
For example, in digital communications, a set of data symbols are represented by a set of unique discrete-time sequences
If one of these sequences has been transmitted, the receiver has to determine which particular sequence has been received by comparing the received signal with every member of possible sequences from the set
§2.8 Correlation of Signals
Similarly, in radar and sonar applications, the received signal reflected from the target is a delayed version of the transmitted signal and by measuring the delay, one can determine the location of the target
The detection problem gets more complicated in practice, as often the received signal is corrupted by additive random noise
§2.8 Correlation of Signals
Definitions A measure of similarity between a pair of en
ergy signals, x[n] and y[n], is given by the cross-correlation sequence rxy [ℓ] defined by
...,,,],[][][ 210
nxy nynxr
The parameter ℓ called lag, indicates the time-shift between the pair of signals
§2.8 Correlation of Signals
y[n] is said to be shifted by ℓ samples to the right with respect to the reference sequence x[n] for positive values of ℓ, and shifted by ℓ samples to the left for negative values of
The ordering of the subscripts xy in the definition of rxy [ℓ] specifies that x[n] is the reference sequence which remains fixed in time while y[n] is being shifted with respect to x[n]
§2.8 Correlation of Signals If y[n] is made the reference signal and shift
x[n] with respect to y[n], then the corresponding cross-correlation sequence is given by
Thus, ryx [ℓ] is obtained by time-reversing rxy [ℓ]
][][][
][][][
xym
nyx
rmxmy
nxnyr
§2.8 Correlation of Signals
The autocorrelation sequence of x[n] is given by
nxx nxnxr ][][][
obtained by setting y[n] = x[n] in the definition of the cross-correlation sequence rxy [ℓ]
Note: , the energy of the signal x[n]
xnxx Enxr
][]0[ 2
§2.8 Correlation of Signals From the relation ryx [ℓ]= rxy [-ℓ] it follows that
rxx [ℓ]= rxx [-ℓ] implying that rxx [ℓ] is an even function for real x[n]
An examination of
reveals that the expression for the cross- correlation looks quite similar to that of the linear convolution
yxyyxxxy EErrr ]0[]0[][
§2.8 Correlation of Signals This similarity is much clearer if we rewrite t
he expression for the cross-correlation as
The cross-correlation of y[n] with the reference signal x[n] can be computed by processing x[n] with an LTI discrete-time
system of impulse response y[−n]
][][)]([][][
yxnynxr
nxy *
x[n] rxy[n]y[-n]
§2.8 Correlation of Signals
Likewise, the autocorrelation of x[n] can be computed by processing x[n] with an LTI discrete-time system of impulse response x[-n]
x[n] rxx[n]x[-n]
Properties of Autocorrelation andCross-correlation Sequences
Consider two finite-energy sequences x[n] and y[n]
The energy of the combined sequence ax[n]+y[n-ℓ] is also finite and nonnegative, i.e.,
0][][][2
][)][][(2
222
nn
nn
nynynxa
nxanynax
Properties of Autocorrelation andCross-correlation Sequences
Thus
a2 rxx[0]+2arxy[ℓ]+ryy[0]≥0
where rxx[0]=Ex>0 and ryy[0]=Ey>0 We can rewrite the equation on the previous
slide as
for any finite value of a
01]0[][
][]0[1
a
rr
rra
yyxy
xyxx
Properties of Autocorrelation andCross-correlation Sequences
or, equivalently,
]0[][
][]0[
yyxy
xyxx
rr
rr
Or, in other words, the matrix
is positive semidefinite 0][]0[]0[ 2 xyyyxx rrr
yxyyxxxy EErrr ]0[]0[][
Properties of Autocorrelation andCross-correlation Sequences
The last inequality on the previous slide provides an upper bound for the cross- correlation samples
If we set y[n] = x[n], then the inequality reduces to
xxxxy Err ]0[][
Properties of Autocorrelation andCross-correlation Sequences
Thus, at zero lag (ℓ=0), the sample valuel of the autocorrelation sequence has its maxi
mum value Now consider the case
y[n] =±bx[n-N] where N is an integer and b > 0 is an arbitrary number
In this case Ey=b2Ex
Properties of Autocorrelation andCross-correlation Sequences
Therefore
xxyx bEEbEE 22
Using the above result in
we get
yxyyxxxy EErrr ]0[]0[][
]0[][]0[ xxxyxx brrbr
Correlation ComputationUsing MATLAB
The cross-correlation and autocorrelation sequences can easily be computed using MATLAB
Example - Consider the two finite-length sequences
x[n]=[1 3 -2 1 2 -1 4 4 2]
y[n]=[2 -1 4 1 -2 3]
Correlation ComputationUsing MATLAB
The cross-correlation sequence rxy[n] computed using Program 2_7 of text is plotted below
Correlation ComputationUsing MATLAB
The autocorrelation sequence rxx[ℓ] computed using Program 2_7 is shown below
Note: At zero lag, rxx[0] is the maximum
Correlation ComputationUsing MATLAB
The plot below shows the cross-correlation of x[n] and y[n]=x[n-N] for N = 4
Note: The peak of the cross-correlation is precisely the value of the delay N
Correlation ComputationUsing MATLAB
The plot below shows the autocorrelation of x[n] corrupted with an additive random noise generated using the function randn
Note: The autocorrelation still exhibits a peak at zero lag
Correlation ComputationUsing MATLAB
The autocorrelation and the cross- correlation can also be computed using the function xcorr
However, the correlation sequences generated using this function are the time- reversed version of those generated using Programs 2_7 and 2_8
Normalized Forms ofCorrelation
Normalized forms of autocorrelation and cross-correlation are given by
They are often used for convenience in comparing and displaying
Note: |ρxx [ℓ]|≤1 and |ρxy [ℓ]|≤1 independent of the range of values of x[n] and y[n]
]0[]0[
][][,
]0[
][][
yyxx
xyxy
xx
xxxx
rr
r
r
r
Correlation Computation forPower Signals
The cross-correlation sequence for a pair of power signals, x[n] and y[n], is defined as
K
KnK
xx nxnxK
r ][][12
1lim][
The autocorrelation sequence of a power signal x[n] is given by
K
KnK
xy nynxK
r ][][12
1lim][
Correlation Computation forPeriodic Signals
1
0~~ ][~][~1
][N
nyx nynxN
r
1
0~~ ][~][~1
][N
nxx nxnxN
r
The autocorrelation sequence of a periodic signal of period N is given by][~ nx
The cross-correlation sequence for a pair of periodic signals of period N, and
is defined as
][~ nx ][~ ny
The periodicity property of the autocorrelation sequence can be exploited to determine the period of a periodic signal that may have been corrupted by an additive random disturbance
Correlation Computation forPeriodic Signals
Note: Both and are also periodic signals with a period N
][~~ yxr][~~ xxr
Correlation Computation forPeriodic Signals
Let be a periodic signal corrupted by the random noise d[n] resulting in the signal
][~ nx
which is observed for 0≤n≤M-1 where M>>N
][][~][ ndnxnw
Correlation Computation forPeriodic Signals
The autocorrelation of w[n] is given by
][][][][
][~][1
][][~1
][][1
][~][~(1
])[][~])([][~(1
][][1
][
~~~~
1
0
1
0
1
0
1
0
1
0
1
0
xddxddxx
M
n
M
n
M
n
M
n
M
n
M
nww
rrrr
nxndM
ndnxM
ndndM
nxnxM
ndnxndnxM
nwnwM
r
Correlation Computation forPeriodic Signals
In the last equation on the previous slide,
is a periodic sequence with a period N and hence will have peaks at ℓ=0, N, 2N,… with the same amplitudes as ℓ approaches M
][~~ xxr
As and d[n] are not correlated, samples of cross-correlation sequences
and are likely to be very small relative to the amplitudes of ][~~ xxr
][~ dxr ][~ xdr
][~ nx
Correlation Computation forPeriodic Signals
The autocorrelation rdd [ℓ] of d[n] will show a peak at ℓ=0 with other samples having rapidly decreasing amplitudes with increasing values of |ℓ|
Hence, peaks of rww [ℓ] for ℓ>0 areessentially due to the peaks of and can be used to determine whether is a periodic sequence and also its period N if the peaks occur at periodic intervals
][~~ xxr][~ nx
Correlation Computation of aPeriodic Signal Using MATLAB
Example - We determine the period of the sinusoidal sequence x[n] =cos(0.25n), 0≤n≤95 corrupted by an additive uniformly distributed random noise of amplitude in the range [-0.5,0.5]
Using Program 2_8 of text we arrive at the plot of rww[ℓ] shown on the next slide
Correlation Computation of aPeriodic Signal Using MATLAB
As can be seen from the plot given above, there is a strong peak at zero lag
However, there are distinct peaks at lags that are multiples of 8 indicating the period of the sinusoidal sequence to be 8 as expected
Correlation Computation of aPeriodic Signal Using MATLAB
Figure below shows the plot of rdd[ℓ]
As can be seen rdd[ℓ] shows a very strongpeak at only zero lag