Speech Signal Representations Part 1: Digital Signal Processing - - PowerPoint PPT Presentation
Speech Signal Representations Part 1: Digital Signal Processing Hsin-min Wang References: 1 X. Huang et al., Spoken Language Processing, Chapters 5-6 2 J. R. Deller et al., Discrete-Time Processing of Speech Signals, Chapters 4-6 3 J. W.
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References:
1 X. Huang et al., Spoken Language Processing, Chapters 5-6 2 J. R. Deller et al., Discrete-Time Processing of Speech Signals, Chapters 4-6 3 J. W. Picone, “Signal modeling techniques in speech recognition,” proceedings of the IEEE, September 1993, pp. 1215-1247
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− A front-end feature extractor (feature extraction module)
recognize
− A back-end classifier (classification module)
theory
3
a
Analog Signal
Digital Signal:
Discrete-time signal with discrete amplitude
Discrete-time Signal or Digital Signal
sampling period=125μs =>sampling rate=8kHz
4
Signal in Signal out
Amplify or attenuate some frequency components of [ ]
Signal in Parameter out Parameter Extraction
2 22 21
m
c c c
2 1
Lm L L
c c c
1 12 11
m
c c c
e.g.:
5
6
− : amplitude (振幅) − : angular frequency (角頻率), − : phase (相角)
[ ]
− = 2 cos π ωn A n x
04 : frequency samples 25 : period . f T = =
1 frequency normalized : ≤ ≤ f f
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− is periodic with period N=8 − is periodic with period N=16 − is not periodic
1
2
3
is not periodic for all values of w
8
[ ] ( )
8 period intergers) are and (both 8 2 4 4 4 cos ) ( 4 cos 4 / cos
1 2 1 1 1 1 1
= ∴ = ⇒ ⋅ = ⇒ + = + = = N k N k N k N N n N n n n x π π π π π π
[ ] ( )
16 period intergers) are and (both 3 16 2 8 3 8 3 8 3 cos ) ( 8 3 cos 8 / 3 cos
2 2 2 2 2 2 2
= ∴ = ⇒ ⋅ = ⇒ + = + = = N k N k N k N N n N n n n x π π π π π π
[ ] ( )
periodic
intergers are and both that condition under the equation this satistify that find t can' 2 ) cos( cos
3 3 3 3 3
⇒ ⋅ = ⇒ + = = k N N k N N n n n x π
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φ
j
( )
φ ω
+
n j
A sinusoidal signal can be expressed as the real part of the corresponding complex exponential
φ
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− taking the real part
( ) ( )
( )
φ ω φ ω φ φ ω φ ω φ ω + + +
n j j n j j j n j n j n j
1 1
1 1
1 1
numbers real are and ,
1
A A A
The sum of N sinusoids of the same frequency is another sinusoid of the same frequency
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12
k
∞ −∞ =
Scale/weighted Time-shifted unit impulse sequence
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− Linear:
− Time-invariant:
n0 samples
2 1 2 1
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k k k
∑ ∑ ∑
∞ −∞ = ∞ −∞ = ∞ −∞ =
k
∑
∞ −∞ =
scale Time-shifted unit impulse sequence linear convolution Time-invariant Time invariant
Digital System
n δ
n h
k n h k n n h n
T T
− → − → δ δ
Impulse response
Unit impulse
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1 2 1 3
Length=M=3
n δ
n h LTI LTI
n x
0 1 2 2 3 1 Length=L=3 Length=L+M-1 3
n δ ⋅ 3
0 1 2 3 9
[ ]
n h ⋅ 3
1 2
1 2 − ⋅ n δ
1 2 3 2 6
[ ]
1 2 − ⋅ n h
2 1
2 1 − ⋅ n δ
2 3 4 1 3
[ ]
2 − n h
Sum up
3 1 11 2 1 3
4
n y
16
k x
k h
k h −
Reflect Multiply Sum up slide
17 0 1 2 1 3
1 3
0 1 2 2 3 1 1
1 3
2 1 1 3
2 1 1 3
3 2 1 3
k h
k x
k h −
1 + −k h
2 + −k h
3 + −k h
4 + −k h
k
∑
∞ −∞ =
Reflect
3 11 1
3 4
, = n n y
1 , = n n y
2 , = n n y
3 , = n n y
4 , = n n y
3 1 11 2 1 3
4
Sum up
n y
1 2 3 4
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[ ] [ ] [ ] [ ] [ ] [ ]
n h n h n x n h n h n x
1 2 2 1
* * * * =
k n x k h k n h k x n x n h n h n x n y
k k
− = − = = =
∑ ∑
∞ −∞ = ∞ −∞ =
* *
– An impulse response has finite duration » Finite-Impulse Response (FIR) – An impulse response has infinite duration » Infinite-Impulse Response (IIR)
[ ]
n h 1
[ ]
n h 2
[ ]
n h 1
[ ]
n h 2
Commutation
[ ] [ ] [ ] ( ) [ ] [ ] [ ] [ ]
n h n x n h n x n h n h n x
2 1 2 1
* * * + = +
[ ]
n h 2
[ ]
n h 1
[ ] [ ]
n h n h
2 1
+
Distribution
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n jw
) ( ω ω ω ω ω j n j k k j n j k k n j
∞ −∞ = − ∞ −∞ = −
∞ −∞ = −
n jwn j
ω
When the input is a complex exponential, the output is another complex exponential of the same frequency and amplitude multiplied by the complex quantity given by
ω j
The discrete-time Fourier transform of h[n]
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− One period can fully describe it, typically –π<w< π − is a complex function of w, it can be expressed as
ω j
( )
ω
ω ω ω ω
j
e H j j j i j r j
∠
magnitude phase
ω j
e H real part imaginary part Cartesian form Polar form
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∑
∞ −∞ = n
( ) ( ) [ ]
m n n m n m m n m n e m n j d e
m n j m n j
− = ≠ = = − − = − =
− − − −
∫
δ π π π ω π
π π ω π π ω
, , 1 sin ) ( 2 1 2 1
) ( ) (
n h m n m h d e m h d e e m h d e e H e n h e H
m m n j m n j m m j n j j n n j j
= − = = = =
∞ −∞ = − − ∞ −∞ = − ∞ −∞ = − − ∞ −∞ = −
δ ω π ω π ω π
π π ω π π ω ω π π ω ω ω ω
2 1 2 1 2 1 : invertible is ansform Fourier tr
) (
∫ ∑
− ∞ −∞ = −
= =
π π ω ω ω ω
ω π d e e H n h e n h e H
n j j n n j j
2 1
absolutely summable DTFT Inverse DTFT
23
− z-transform of h[n] is defined as
∑
∞ −∞ = −
n n
z H n h
ω j
e H n h
ω j
ω
ω
j
e z j
z H e H
=
=
jω
complex plane
unit circle Im Re
ω j
frequency response
filter characteristics, e.g. stability
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− In general, ROC is a ring-shaped region and the Fourier transform exists if its ROC includes the unit cycle
− Necessary and sufficient condition:
− ∞ −∞ =
∑
n n
complex plane
R1 R2 Re Im
∑
∞ −∞ = n
A sufficient condition for the existence of Fourier transform
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− Similarly, an LTI system is anti-causal if
All physical systems are causal
26
1
n
1 1 1
− ∞ = − − ∞ =
n n n n n
If
1 < −
have a pole at z=a (Pole: z-transform goes to infinity)
1
Re Im
1
the unit cycle
A causal and stable system has all its poles inside unit circle
27
2
n
1 1 1 1 1 1 1 2
− − − − ∞ = − ∞ = − − − −∞ = − ∞ −∞ =
∑ ∑ ∑ ∑
n n n n n n n n n n n
If
1
− z
2
Re Im
−∞ → < n n h n h a as lly exponentia go will because exist, t doesn'
ansform Fourier tr the , 1 when
2 2
the unit cycle
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3
n n
− + − = − + + =
− −
2 1 3 1 12 1 2 2 1 1 1 3 1 1 1
1 1 3
z z z z z z z H
[ ] [ ]
2 1 , 2 1 1 1 1 2 1 3 1 , 3 1 1 1 3 1
1 1
< − → ← − − − > + → ← −
− −
z z n u z z n u
z n z n
Re Im
the unit cycle
2 1 3 1 −
circle unit the include t doesn' because exist, t doesn'
ansform Fourier tr
3 3
ROC n h
3
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4
n
N N N N N n n N n n n
− − − − = − − = −
∑ ∑
1 1 1 1 1 1 4
1 3 2 2 1
.....
− − − −
+ + + +
N N N N
a z a az z
4
Im
the unit cycle
3 1 −
7 poles at zero A pole and zero at is cancelled
4 π
( )
2
N k j k π
If N=8
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{ } { } { } { } { } { }
{ } { }
{ }
{ }
[ ] ( ) { }
{ }
( ) { }
) ( ] [ ] [ ] [ ) ( ] [ ] [ ] [ ) ( Im ) ( ) ( 2 1 2 / ]) [ ] [ ( ] [ Odd ) ( Re ) ( ) ( 2 1 ] [ ] [ 2 1 2 / ]) [ ] [ ( ] [ Even ) ( ) ( if
and if
is ) ( ) ( ) ( if
and if even is ) (
is ) ( arg even, is ) ( ) ( ) ( ) (
is ) ( Im even, is ) ( Re ) ( Im ) ( Re ) ( Im ) ( Re ) ( ) ( ) ( real ] [ ) ( )] ]( [ [ ) ]( [ ] [ ) ( )] ]( [ [ ) ]( [ ] [ ] [ ) ( ] [ ] [ ] [ ] [ ) ( ] [
) ( ) ( ) ( ) ( arg ) ( arg * * * * * * ) ( * * * * ) ( * ) ( * * * ) )( ( w w j n n w w j n jwn n jw n jw jwn jw jwn n n n jw n jwn jw jw jw jw jw jw n jwn n jwn jw jw e X j jw e X j jw jw jw jw jw jw jw jw jw jw jw jw n jwn n n jw jw n n w j n n w j n jwn jw n n w j n jwn n jwn jw
e X e n x e e n x e n x e e X e e n n x e n n x n n x e X j e X e X n x n x n x e X e X e X e n x e n x n x n x n x x f x f x f x f x f x f e X e X e e X e e X e X e X e X e X j e X e X j e X e X e X e X n x e X e n x e n x n x e X e n x e n x e n x n x e X e n x e n x n x e n x e X n x
jw jw
− ∞ −∞ = − − ∞ −∞ = − − − ∞ −∞ = − − ∞ −∞ = − − − ∞ −∞ = − ∞ −∞ = − − − − − − ∞ −∞ = − ∞ −∞ = − − − ∞ −∞ = − − ∞ −∞ = − − ∞ −∞ = − − ∞ −∞ = − − − ∞ −∞ = − ∞ −∞ = −
= ∑ = ∑ ↔ = ∑ − = ∑ − ↔ − = − ↔ − − = = + = ∑ − + ∑ ↔ − + = − = − = − ⇒ = = ⇒ + = − = ⇔ = ⇔ = ∑ = ∑ − ↔ − = ∑ = ∑ = ∑ ↔ = ∑ − = ∑ − ↔ − ∑ = ↔
−
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] [ ] [ ] [ * z H z X z Y n h n x n y = ↔ =
) ( ) ( ) ( ] [ ) (
) (
z H z X z H z k x z z k n h k x z k n h k x z n y z Y
k k k k k n n n n k n n
= = − = − = =
∞ −∞ = − ∞ −∞ = − − − ∞ −∞ = − ∞ −∞ = ∞ −∞ = ∞ −∞ = −
The z-transform of the convolution of two signals is the product of their z-transforms
jw jw jwn n k jwk k jwk k n jw n jwn n k jw
− ∞ −∞ = ∞ −∞ = − ∞ −∞ = − − − ∞ −∞ = − ∞ −∞ = ∞ −∞ = ) (
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
jw jw jw jw jw jw jw jw jw
e H e X e Y e H e X e Y e H e X e Y ∠ + ∠ = ∠ = ⇒ = ,
jw jw jw
33
jw jw
) ( ) ( ) (
n j j n j n w j w j j jwn j j jwn jw jw
− − − − − − − − −
τ π π τ τ τ π π τ π π τ π π τ ω π π τ π π
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We can compute the signal’s energy in the time domain or in the frequency domain.
− ∞ −∞ = π π ω
j n 2 2
power spectrum
* *
l m xx
∗ ∞ −∞ = ∞ −∞ =
2 *
xx F xx
− − π π ω π π ω
n j n j xx xx 2
∞ −∞ = − ∞ −∞ =
n n xx
π π
2 2 *
The autocorrelation of signal x[n]
35
− The output of a LTI system to a complex exponential is another complex exponential
quantity being their eigenvalue
n j k j n j k n j
ω ω ω ω
− ∞ ∞ = − ∞ ∞ =
∑ ∑
) (
.
s system' the as to referred
is It response. impulse system the
ansform Fourier tr the : unction transfer f response frequency ω H
jw
e H
n j
ω
36
n jw j n jw j
e e A e e A n w A n x 2 2 cos
− −
+ = + =
φ φ
φ
( )
( )
) ( ) ( ) ( ) (
n j ω H j n j ω H j n j * n j n j j n j j
+ − ∠ − + ∠ + − + − −
φ ω φ ω φ ω φ ω ω φ ω φ
If |H(w0)|>1, the LTI system will amplify that frequency. If |H(w0)|<1, it will attenuate it.
37
− Speech signals can be decomposed as sums of sinusoids − A similar expression is obtained for an input consisting of a sum
=
K k k k k
1
=
K k k k k k k
1
38
39
/ 2 1 / 2 1
− = − − =
N kn j N k N N N kn j N n N N π π
Inverse DFT, Synthesis DFT, Analysis
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
( )
[ ] [ ]
( )
[ ]
( )
[ ] [ ] [ ]
∑ ∠ + + = ∑ + + = ∑ + − + = = − ∑ =
= = = − − = 18 1 18 1 / 2 * / 2 18 1 / 2 / ) ( 2 * / 2 18 18
) / 2 cos( 2 1 1 1 ~
k N N N k N kn j N N kn j N N k N kn j N N n k j N N N N N kn j k N N
k X N kn k X N N X e k X e k X N N X e k X e k X N N X k X k X e k X N n x π
π π π π π
40
[n] ) ( lim ) ( , 1 ) ( / 1 ) (
δ w d w dw w d
w w d
∆ → ∆ ∞ ∞ ∆ ∆
≡ > ∆ ∀ ∫ = ⇒ ∆ < ≤ ∆ ≡
The Dirac delta The Kronecker delta
∆ 1
w
∆ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
w X du u w w u X w w w X w w w X w w w X − ∫ = − − = − ∗ − = −
∞ ∞
δ δ δ δ
n jw jwn jwn
∞ ∞
∞ ∞ ∞ ∞ ∆ → ∆ ∞ ∞ ∆ → ∆ ∞ ∞
) ( 2 ) ( 2 2 1
w e e dw w w e
n jw n jw jwn
− ↔ ∴ ∫ = −
∞ ∞
πδ πδ π
∫ =
− π π ω ω
ω π d e e H n h
n j j
) ( 2 1 ] [
41
∞ −∞ = k N
− = − = − = − 1 / 2 1 / 2 1 / 2
N k N nk j N k N nk j N N N n N nk j N N
π π π
The impulse train is periodic with period N Alternate expression of an impulse train
n jw
− = 1
N k jw N
The Fourier transform of an impulse train signal is also an impulse train
42
∑
∞ −∞ =
k
∞ −∞ =
k n
DTFT
− =
∑
∞ −∞ =
k P P X
k
π ω δ π ω 2 2
jw
ae H
−
− = 1 1 ω
DTFT
jw jw
e X e H n h n x ⇔ ∗
( ) ( ) ( )
− − = − − = =
∞ −∞ = − ∞ −∞ = −
k P ae P k P P ae X H Y
k k P j k j
π ω δ π π ω δ π ω ω ω
π ω
2 1 1 2 2 2 1 1
2
43
∑ − = ∑ − =
− = − = 1 / 2 1
) / 2 ( ) ( 2 ) / 2 ( 2 ) ( ) (
N k N k j N k jw jw N
N k w e X N N k w N e X e X π δ π π δ π
π
N k k N
∞ −∞ = ∞ −∞ =
N
Given a periodic signal xN[n] with period N, Then
44
45
< =
w w e H
jw
| | 1 ) (
It lets all frequencies below w0 pass through unaffected and completely blocks frequencies above w0
) 2 sinc( ) sin( 2 ) sin( 2 2 ) ( 2 1 ] [ n f n n jn n j jn e e d e n h
n j n j w w n j
π π ω π ω π ω π ω π
ω ω ω
= = = − = ∫ =
− −
x x x π π ) sin( ) sinc( ≡
46
2 / ) 1 ( 2 / ) 1 ( 2 / 2 / 2 / 2 / 2 / 2 / 1 1
) ( ) 2 / sin( ) 2 / sin( 1 1 ) ( 1 1 ) ( ] [ ] [ ] [
− − − − − − − − − − − − − = −
= = − − = − − = − − = ∑ = − − ≡
N jw N jw jw jw jw jwN jwN jwN jw jwN jw N N n n
e w A e w wN e e e e e e e e e H z z z z H N n u n u n h
π π π
…
N-1
n 1
(w/2π)
N w π 2 =
0.02
,...} 2 , , { / 2 for , ) ( N N k N k w w A
k
± ± ≠ = = π
47
The generalized Hamming window is defined as
) / 2 cos( ) 1 ( ] [ < ≤ − − ≡
N n N n n hh π α α
) ( ) 2 / ( ) ( ) 2 / ( ) ( ) 1 ( ) ( ] [ ) 2 / ( ] [ ) 2 / ( ] [ ) 1 ( ) / 2 cos( ] [ ] [ ) 1 ( ] [
) / 2 ( ) / 2 ( / 2 / 2 N w j N w j jw jw h N n j N n j h
e H e H e H e H e n h e n h n h N n n h n h n h
π π π π π π π π π π π π
α α α α α α π α α
+ − −
− − − = − − − = − − =
31 dB 44 dB 31 dB 44 dB
Decay with frequency rapidly Constant for all frequencies
N π 4 N π 4
window Hamming the 46 . window Hanning the 5 . ⇒ = ⇒ = α α
48
∑
∞ −∞ =
k
− = − − =
1 ,......, 1 , , 1 2 cos 46 . 54 . N n N n n w π
Hamming window
jw jw
e X e W n w n x ∗ ⇔ π 2 1 ( ) ( ) ( ) ( ) ( ) ( )
∑ − = ∑ ∫ − − = ∑ − ∗ = − ∑ ∗ = ∗ =
∞ −∞ = ∞ −∞ = ∞ ∞ − ∞ −∞ = ∞ −∞ = k k k k
k P W P d k P W P k P W P k P P W X W Y π ω τ τ π ω δ τ π ω δ ω π ω δ π ω π ω ω π ω 2 1 2 1 2 1 2 2 2 1 2 1
49
− The impulse response of an FIR filter has finite duration − Have no denominator in the rational function H(z)
− Can be implemented with a simple train of delay, multiple, and add operations − FIR systems are always stable
= = M r r M r
z-1 z-1 z-1 M
b
1
b b
n y
n x
≤ ≤ =
, , M n b n h
n
∑ = =
= − M r r r z
b z X z Y z H
finite is ∑
= M r r
b Q
50
phase linear be to ] [ for condition sufficient a is ] [ ] [ system a called is ] [ so ,
function linear a , ) (
function even and real a is ) ( so cosines,
n combinatio linear a is ) ( ) ( ) cos( ] [ 2 ] [ )) ( cos( ] [ 2 ] [ ] [ ] [ ] [ ] [ ] [ ] [ ) (
* 1 1 1 ) ( ) ( 1 ) 2 (
n h n M h n h se linear-pha n h w Lw e H w w A w A e w A e wn L n h L h e L n w n h L h e e e n h e L h e n M h e n h e L h e n h e H
jw jwL jwL L n jwL L n jwL L n L n jw L n jw jwL L n n L jw jwn jwL M n jwn jw
− ± = − = ∠ = ∑ + + = ∑ − + = ∑ + + = ∑ − + + = ∑ =
− − = − − = − − = − − − − − = − − − − = − 2L h[L] h[2L] h[0] L
L-1 -w1 0 -wL L+L wL L+1 w cos(w)=cos(-w) h[L+1]=h[L-1]
51
1
−
jw jw
−
+ − = + = + + = − + = ω α ω α θ ω α ω α ω α ω ω α
ω ω
cos 1 sin arctan cos 2 1 sin cos 1 sin cos 1
2 2 2 2 j j
e j e H
2
log 10
ω j
e H
α>0: low-pass filter α<0: high-pass filter α controls the slope of the curve
52
= =
N k M k k k
1
k N k M k k k k
z z X b z z Y a z Y
− = = −
+ =
1
= − = −
− = =
N k k k M k k k
z a z b z X z Y z H
1
1
delay property
n
z z X n n x z X n x
−
→ − →
linearity and delay properties A rational transfer function
z-1 z-1 z-1 M
b
2
b b
n y
n x
1
b
z-1 z-1 z-1 1
a
2
a
N
a
( )
= − − = − −
− − =
N k k L M r r L
z d z c Az z H
1 1 1 1
1 1 zeros poles
53
= −
− =
N k k k
z d A z H
1 1
1
If M<N
− = − = −
+ − =
N M k k k N k k k
z B z d A z H
1 1
1
If M≥N
causal and stable both is ] [ 1 ] [ ] [ 1 If stable is ] [ 1 ] 1 [ ] [ 1 If 1 ] [ ] [ 1 If
1 1 1
n h z d A z H n u d A n h d n h z d A z H n u d A n h d z d A z H n u d A n h d
k k k k n k k k k k k k k n k k k k k k k n k k k k − − −
− = ↔ = < − = ↔ − − − = > − = ↔ = <
=
N k n k k n
1
has an infinite impulse response IIR is not guaranteed to be stable and causal like FIR, IIR do not have linear phase, IIR is more efficient than FIR in realizing steeper roll-offs with fewer coefficients.
54
55
w A e H e A e H n u A n h z A z H n y n Ax n y
jw jw jw n
cos 2 1 | | | ) ( | is square magnitude The 1 ) ( is ansform Fourier tr The ] [ ] [ is response impulse The 1 ) ( is function transfer The ] 1 [ ] [ ] [
2 2 2 1
α α α α α α − + = ⇒ − = ⇒ = ⇒ − = ⇒ − + =
− −
be both stable and causal is |α|<1
56
2 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 2 2 1 1 2 1
) cos( 2 1 ) 1 )( 1 ( ) ( conjugate complex are roots two 4 If : II Case systems
two to degenerate roots real two 4 If : I Case 2 4 by given are poles two the real, are and , , If zeros. no and poles two has system This 1 ) ( is function transfer The ] 2 [ ] 1 [ ] [ ] [
− − − − − − − − + − ± − − −
+ − = − − = ⇒ = ⇒ ⇒ < + ⇒ ⇒ > + + ± = ⇒ − − = ⇒ − + − + = z e z w e A z e z e A z H e z a a a a a a a z a a A z a z a A z H n y a n y a n Ax n y
jw jw jw σ σ σ σ σ
A bandpass filter it favors frequencies in a band around w0
57
58
The Fourier transform of an analog signal x(t) is defined as
dt e t x X
t j
∞ ∞ − Ω −
= Ω ) ( ) (
with its inverse transform being
Ω Ω =
∞ ∞ − Ω d
e X t x
t j
) ( 2 1 ) ( π dt e t x X
t j
∞ ∞ − Ω −
= Ω ) ( ) ( Ω Ω =
∞ ∞ − Ω d
e X t x
t j
) ( 2 1 ) ( π
∞ −∞ = −
=
n jwn jw
e n x e X ] [ ) ( dw e e X n x
jwt jw
−
=
π π
π ) ( 2 1 ] [
Discrete time Fourier transform of x[n]
59
Impulse Train To Sequence
∞ −∞ =
− =
n
nT t t p δ
Sampling
∞ −∞ = ∞ −∞ = ∞ −∞ =
− = − = − = =
n n n p
nT t n x nT t nT x nT t t x t p t x t x ) ( ] [ ) ( ) ( ) ( ) ( ) ( ) ( ) ( δ δ δ
Continuous-Time to Discrete-Time Conversion Periodic Impulse Train Discrete-Time Signal Continuous-Time Signal Digital Signal
) (t x
∞ −∞ =
− =
n
nT t t p ) ( ) ( δ
∫ = ≠ ∀ =
∞ ∞ −
1 ) ( , ) ( dt t t t δ δ
Discrete-time signal with discrete amplitude
60
( )
Ω j X p
( )
Ω j X p
( )
Ω j P
frequency) (sampling 2 2
s s
F T π π = = Ω
= Ω < Ω T
s N
π 2 1 = Ω > Ω T
s N
π 2 1
aliasing distortion ( ) ( )
∞ −∞ =
Ω − Ω = Ω
k s
k T j P δ π 2
( ) ( ) ( ) ( ) ( ) ( )
∞ −∞ =
Ω − Ω = Ω Ω ∗ Ω = Ω
k s p p
k j X T j X j P j X j X 1 2 1 π
frequency the : 2 / Nyquist Fs
N
Ω −
( )
Ω j X
N
Ω
Ω