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Young Won Lim10/28/16
Spectrum Representation (5A)
Young Won Lim10/28/16
Copyright (c) 2009 - 2016 Young W. Lim.
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5A Spectrum Representation
3 Young Won Lim10/28/16
Fourier Series with real coefficients
x (t ) = a0 + ∑k=1
∞
(ak cos(kω0 t) + bk sin(kω0 t ))
x(t ) = g0 + ∑k=1
∞
gk cos(kω0 t + ϕk)
x (t ) = X 0 + ∑k=1
∞
ℜ{X k e+ j kω0 t}
1
a
b
X k = gk⋅e+ jϕk
5A Spectrum Representation
4 Young Won Lim10/28/16
Phasor Representation
x(t ) = g0 + ∑k=1
∞
gk cos(kω0 t + ϕk)
x (t ) = g0 + ∑k=1
∞
gk ℜ{e+ j (kω0 t + ϕk)}
x (t ) = g0 + ∑k=1
∞
ℜ{gk⋅e+ jϕk⋅e+ j kω0 t}
X k via gk , ϕk
X k = gk⋅e+ jϕk
5A Spectrum Representation
5 Young Won Lim10/28/16
Fourier Series with complex coefficients
x (t ) = ∑k=−∞
+∞
C k e+ j kω0 t
x(t ) = g0 + ∑k=1
∞
gk cos(kω0 t + ϕk)
x (t ) = X 0 + ∑k=1
∞
ℜ{X k e+ j kω0 t}
4
a
b
Ck =12g+k e
+ jϕk (k > 0) C k =12g−k e− jϕk (k < 0)
5A Spectrum Representation
6 Young Won Lim10/28/16
Complex Coefficient
x (t ) = ∑k=−∞
+∞
C k e+ j kω0 t
4
Ck =12g+k e
+ jϕk (k > 0) C k =12g−k e− jϕk (k < 0)
= g0 + ∑k=1
∞
gk⋅12
(e+ j(kω0 t + ϕk) + e− j(kω0 t + ϕk))
= g0 + ∑k=1
∞
([ 12 gk e+ jϕk ] e+ j k ω0 t + [ 12 gk e− jϕk ]e− j kω0 t)= g0 + ∑
k=1
∞
( [C k ] e+ j kω0 t + [C−k ]e
− j kω0 t )
Ck via gk , ϕk
x (t) = g0 + ∑k=1
∞
gk⋅cos(kω0 t + ϕk)
⋯ , C−1 , C0 , C+1 , ⋯
g0 , g1 , g2 , ⋯
5A Spectrum Representation
7 Young Won Lim10/28/16
Fourier Coefficients Relationship
a0 =1T ∫0
Tx (t) dt
ak =2T ∫0
Tx (t) cos (kω0 t) dt
bk =2T ∫0
Tx (t) sin (kω0 t) dt
g0 = a0
gk = √ak2 + bk
2
ϕk = tan−1 (−bk
ak)
X 0 = g0
X k = gk⋅e+ jϕk
1 a
b
k = 1, 2, .. .
Ck = 12 g+k e
+ jϕk (k > 0)
C k = 12 g−k e
− jϕk (k < 0)
4
k = 1, 2, .. .
k = 1, 2, .. .k = 0, ±1, ±2, .. .
5A Spectrum Representation
8 Young Won Lim10/28/16
Single-Sided Spectrum
x(t) = a0 + ∑k=1
∞
(ak cos(k ω0 t) + bk sin(k ω0 t)) x(t) = g0 + ∑k=1
∞
g k cos(k ω0 t + ϕk )
a0 =1T ∫0
Tx( t) dt
a k =2T∫0
Tx (t) cos (k ω0 t) dt
bk =2T∫0
Tx (t ) sin (k ω0 t) dt
g0 = a0
gk = √ ak2+ bk
2
ϕk = tan−1 (−bk
ak)
a0 , a1 , a2 , ...b0 , b1 , b2 , ...
g0 , g1 , g2 , ...ϕ0 , ϕ1 , ϕ2 , ...
5A Spectrum Representation
9 Young Won Lim10/28/16
Single-Sided Spectrum
x(t) = a0 + ∑k=1
∞
(ak cos(k ω0 t) + bk sin(k ω0 t)) x(t) = g0 + ∑k=1
∞
g k cos(k ω0 t + ϕk )
= cos (α) cos(β)
gk cos(k ω0 t + ϕk ) = gk cos (ϕk ) cos(k ω0 t) − gk sin (ϕk) sin (k ω0 t)
cos(α+β) − sin (α) sin(β)
ak cos(kω0 t ) + bk sin(kω0 t )
a k = g k cos(ϕk)
−bk = g k sin (ϕk )
g0 = a0
gk = √ ak2+ bk
2
ϕk = tan−1 (−bk
ak)
a0 = g0
a k2+ bk
2= g k
2
−bk
ak
= tan (ϕk)
5A Spectrum Representation
10 Young Won Lim10/28/16
Periodogram
x(t ) = g0 + ∑k=1
∞
gk cos(kω0 t + ϕk)
a
Periodogram
|gk| = √ak2 + bk
2
One-Sided
g0 , g1 , g2 , ⋯
5A Spectrum Representation
11 Young Won Lim10/28/16
Power Spectrum
x (t ) = ∑k=−∞
+∞
C k e+ j kω0 t
4
Power Spectrum
|C k|2 = 1
4 (ak2 + bk
2)
Two-Sided
⋯, C−1 , C0 , C+1 , ⋯
5A Spectrum Representation
12 Young Won Lim10/28/16
Power Spectrum
x (t ) = ∑k=−∞
+∞
C k e+ j kω0 t
4
Power Spectrum
|C k|2 = 1
4 (ak2 + bk
2)
P =1T∫T
|x (t )|2dt
=1T∫T
x (t)x∗(t )dt
=1T∫T
x∗(t) ∑
k=−∞
+∞
Ck e+ j kω0 t dt
=1T
∑k=−∞
+∞
C k∫T
x∗(t) e+ j kω0 t dtC k =
1T∫0
T
x (t) e− j kω0 t dt
C k∗
=1T∫0
T
x∗(t) e
+ j kω0 t dt =1T
∑k=−∞
+∞
C kT C k∗
=|C k|2
5A Spectrum Representation
13 Young Won Lim10/28/16
Two-Sided Spectrum
x ( t) = g0 + ∑k=1
∞
gk cos(kω0 t + ϕk)x (t ) = a0 + ∑k=1
∞
(ak cos(k ω0 t ) + bk sin(kω0 t ))
x ( t) = ∑k=−∞
+∞
C k e+ j kω0 t
Ck =
a0 (k = 0)12 (ak − j bk) (k > 0)12 (ak + j bk) (k < 0)
|C k| =a0 (k = 0)12 √ak
2 + bk2 (k ≠ 0)
Arg(C k ) =tan−1 (−bk /ak ) (k > 0)
tan−1 (+bk /ak ) (k < 0)
x (t) = ∑k=−∞
+∞
C k e+ j kω0 t
Ck =
g0 (k = 0)12 g+k e
+ jϕk (k > 0)12 g−k e− j ϕk (k < 0)
|C k| =g0 (k = 0)12|gk| (k ≠ 0)
Arg(C k ) = +ϕk (k > 0)
−ϕk (k < 0)
Power Spectrum Periodogram
|C k|2 = |C−k|
2 = 14|gk|
2 = 14 (ak
2 + bk2) |gk| = 2⋅|C k| = √ak
2 + bk2
Two-Sided One-Sided
5A Spectrum Representation
14 Young Won Lim10/28/16
Spectrum
Real Single-tone Sinusoidal Signal
x(t) = A cos(ω0 t + ϕ)
= ℜ{X e jω0 t}
X = A e jϕ
=A2
(e+ j(ω0 t + ϕ)+ e− j(ω0 t + ϕ)
)
=A2
(e+ jϕe+ jω0 t + e− jϕe− jω0 t)
={A e+ jϕ
}
2e+ jω0 t +
{A e+ jϕ}
2
∗
e− jω0 t= {X2 e+ jω0t +X∗
2e− jω0t }
A cos(ω0 t + ϕ)
5A Spectrum Representation
15 Young Won Lim10/28/16
Spectrum
x(t) = A0 + ∑k=1
N
A k cos(ωk t + ϕk)
Real Multi-tone Sinusoidal Signal
= X 0 + ℜ{∑k=1
N
X k ejωk t }
X 0 = A0
= X 0 + ∑k=1
N
{X k
2e+ jωk t +
X k∗
2e− jωk t}
the phasor of angular frequency ωk
X k = A k ejϕk
Real Single-tone Sinusoidal Signal
x(t) = A cos(ω0 t + ϕ)
= ℜ{X e jω0 t}
X = A e jϕ
= {X2 e+ jω0t +X∗
2e− jω0t }
5A Spectrum Representation
16 Young Won Lim10/28/16
Spectrum
only magnitude is display
X k
2
X k∗
2
X k
2e+ jωk t
X k∗
2e− jωk t
−ωk +ωk
x(t) = A0 + ∑k=1
N
A k cos(ωk t + ϕk)
Real Multi-tone Sinusoidal Signal
= X 0 + ℜ{∑k=1
N
X k ejωk t }
X 0 = A0
= X 0 + ∑k=1
N
{X k
2e+ jωk t +
X k∗
2e− jωk t}
the phasor of angular frequency ωk
X k = A k ejϕk
5A Spectrum Representation
17 Young Won Lim10/28/16
Frequency Resolution of Fourier Series Representations
+∞−∞
+π−π
ω̂0 ω̂0
+∞0
0
0
0
T s T s
CTFS
DTFS / DFT
ω0 ω0
ω0 =2π
T 0
ω̂0 =2π
N 0
period : T 0 seconds
period : N0 samples
Continuous Time
Discrete Time
Discrete Frequency
Normalized Discrete Frequency
Fourier Analysis Overview (0B) 18 Young Won Lim
10/28/16
Sampling Period and the Number of Samples
x (t )
T 0 period
N 0 samplesT s
x [n]
ω0 =2π
N0T s
ω0 =2π
T 0
ω0
1/T s
=2πN 0T s
T s
ω̂0 =2π
N 0
normalized frequency resolution
frequency resolution
T 0 = N 0T s
⋅T s ⋅ f s
Fourier Analysis Overview (0B) 19 Young Won Lim
10/28/16
Sampling Period and the Number of Samples
x (t )
T 0 period
N 1 samples
N 2 samples
N 1 < N2
T 1 > T 2
x1[n]
x2[n]
T 1N1 = T 2N2
ωs =2π
T s
2πT 1
2πT2
ω1
ω2
ωs = ω1
ωs = ω2
ω0T 1 > ω0T 2
T s = T 1
T s = T 2
ω1 < ω2
ω0 =ω̂0
T s
5A Spectrum Representation
20 Young Won Lim10/28/16
Frequency and Digital Frequency
ω̂ (rad /sample)
ω (rad /sec )
Frequency
Digital Frequency
−ωs +ωs−ω0 +ω0−ω0−ωs +ω0+ωs
−ω̂0 +ω̂0 +ω̂0+2π−ω̂0−2π +2π−2π
e− jω s t e+ jωs te− jω 0 t e+ jω0 te− j (ω0+ω s)t e+ j(ω0+ωs) t
e− j ω̂ 0n e+ j ω̂0 n e+ j( ω̂0+2π )ne− j (ω̂0−2π )n e+ j+2πe− j2π n
5A Spectrum Representation
21 Young Won Lim10/28/16
Frequency and Digital Frequency
ω̂ = ω⋅T s
ω̂ = ωf s
tT s
T 0
x(t )= cos(ω0 t) x [n] = x(nT s)= cos(nω0T s)
5A Spectrum Representation
22 Young Won Lim10/28/16
γ[k ] =1N
∑n = 0
N−1
x [n] e− j k ω̂0n x [n] = ∑k = 0
N−1
γ[k ] e+ j k ω̂0n
Fourier Transform Types
Continuous Time Fourier Series
C k =1T∫0
Tx (t ) e− j k ω0 t dt
Discrete Time Fourier Series
x (t ) =12π
∫−∞
+∞
X ( jω) e+ jω t dω
Continuous Time Fourier Transform
x [n] =12π
∫−π
+π
X ( j ω̂) e+ j ω̂nd ω̂
Discrete Time Fourier Transform
X ( j ω̂) = ∑n =−∞
+∞
x [n] e− j ω̂n
x (t ) = ∑k=−∞
+∞
Ck e+ j kω0 t
X ( jω) = ∫−∞
+∞
x (t) e− jω t dt
5A Spectrum Representation
23 Young Won Lim10/28/16
Continuous Time
Continuous Time Fourier Series
Continuous Time Fourier Transform
C k =1T∫0
Tx (t ) e− j k ω0 t dt x (t ) = ∑
k=−∞
+∞
Ck e+ j kω0 t
x (t ) =12π
∫−∞
+∞
X ( jω) e+ jω t dωX ( jω) = ∫−∞
+∞
x (t) e− jω t dt
5A Spectrum Representation
24 Young Won Lim10/28/16
Discrete Time
Discrete Time Fourier Series
Discrete Time Fourier Transform
γ[k ] =1N
∑n = 0
N−1
x [n] e− j k ω̂0n x [n] = ∑k = 0
N−1
γ[k ] e+ j k ω̂0n
x [n] =12π
∫−π
+π
X ( j ω̂) e+ j ω̂nd ω̂X ( j ω̂) = ∑n =−∞
+∞
x [n] e− j ω̂n
5A Spectrum Representation
25 Young Won Lim10/28/16
Continuous Time Signal Spectrum
CTFS
only magnitude is display
C kC−k
+ k ω0−k ω0
C k e+ j kω0 t
CTFT
only magnitude is display
X (+ jω)X (− jω)
+ ω−ω
X (+ jω)e+ jω tX (− jω)e− jω tC−k e− j kω 0 t
C k =1T∫0
Tx (t ) e− j k ω0 t dt X ( jω) = ∫
−∞
+∞
x (t) e− jω t dt
5A Spectrum Representation
26 Young Won Lim10/28/16
Discrete Time Signal Spectrum
DTFT
only magnitude is display
X (+ j ω̂)X (− j ω̂)
+ ω̂−ω̂
X (e+ j ω̂)e+ j ω̂ nX (e− j ω̂)e− j ω̂n
DTFS
only mag is display
γ [+k ]γ [−k ]
+k ω̂0−k ω̂0
γ [+k ] e+ jk ω̂0nγ [−k ]e− j k ω̂0n
+ π−π+ π−π
γ[k ] =1N
∑n = 0
N−1
x [n] e− j k ω̂0n X ( j ω̂) = ∑n =−∞
+∞
x [n] e− j ω̂n
5A Spectrum Representation
27 Young Won Lim10/28/16
Young Won Lim10/28/16
References
[1] http://en.wikipedia.org/[2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003[3] M.J. Roberts, Fundamentals of Signals and Systems[4] S.J. Orfanidis, Introduction to Signal Processing[5] K. Shin, et al., Fundamentals of Signal Processing for Sound and Vibration Engineerings
[6] A “graphical interpretation” of the DFT and FFT, by Steve Mann