[PPT] - Digital Signal Processing Berlin Chen 2004 References: 1. X. Huang PowerPoint Presentation

SLIDE 1

Digital Signal Processing

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

Berlin Chen 2004

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Analog Signal to Digital Signal

[ ]

( )

period sampling : , T nT x n x

a

=

rate sampling 1T Fs =

Analog Signal Discrete-time Signal or Digital Signal

nT t =

sampling period=125μs =>sampling rate=8kHz

Digital Signal:

Discrete-time signal with discrete amplitude

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Analog Signal to Digital Signal (cont.)

Impulse Train To Sequence

[ ] ( ))

( ˆ nT x n x

a

= Sampling

( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( )

∑ ∑ ∑

∞ −∞ = ∞ −∞ = ∞ −∞ =

− = − = − = =

n n a n a a s

nT t n x nT t nT x nT t t x t s t x t x δ δ δ

( )

t x a

Continuous-Time Signal Discrete-Time Signal Digital Signal Discrete-time signal with discrete amplitude

[ ]

n x

Continuous-Time to Discrete-Time Conversion

( ) ( )

∑

∞ −∞ =

− =

n

nT t t s δ

Periodic Impulse Train ( ) [ ]

n x t xs by specified uniquely be can switch

( )

t x a

( ) ( )

∑

∞ −∞ =

− =

n

nT t t s δ

( ) ( )

∫

∞ ∞ −

= ≠ ∀ = 1 , dt t t t δ δ

1 T 2T 3T 4T 5T 6T 7T 8T

T
2T

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Analog Signal to Digital Signal (cont.)

A continuous signal sampled at different periods

T 1 ( )

t x a

( )

t x a

( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( )

∑ ∑ ∑

∞ −∞ = ∞ −∞ = ∞ −∞ =

− = − = − = =

n n a n a a s

nT t n x nT t nT x nT t t x t s t x t x δ δ δ

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Analog Signal to Digital Signal (cont.)

( )

Ω j Xa

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 S δ π 2

( ) ( ) ( ) ( ) ( ) ( )

∑

∞ −∞ =

Ω − Ω = Ω Ω ∗ Ω = Ω

k s a s a s

k j X T j X j S j X j X 1 2 1 π

N

Ω

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Ω > Ω ⇒ Ω > Ω − Ω 2

N s N N s

Q

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Ω < Ω ⇒ Ω < Ω − Ω 2

N s N N s

Q

( )

⎩ ⎨ ⎧ Ω < Ω = Ω

Ω

therwise

2 /

s

T j R

s

( ) ( ) ( )

Ω Ω = Ω

Ω

j X j R j X

p a

s

( ) ( )

Ω Ω j X j X

p a

from recovered be t can'

Low-pass filter high frequency components got superimposed on low frequency components

Spectra

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Analog Signal to Digital Signal (cont.)

To avoid aliasing (overlapping, fold over)

– The sampling frequency should be greater than two times of frequency of the signal to be sampled → – (Nyquist) sampling theorem

To reconstruct the original continuous signal

– Filtered with a low pass filter with band limit

Convolved in time domain

s

Ω

( )

t t h

s

Ω = sinc

( ) ( ) ( ) ( ) ( )

∑ ∑

∞ −∞ = ∞ −∞ =

− Ω = − =

n s a n a a

nT t nT x nT t h nT x t x sinc

N s

Ω > Ω 2

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Two Main Approaches to Digital Signal Processing

Filtering
Parameter Extraction

Filter

Signal in Signal out

[ ]

n x

[ ]

n y

Parameter Extraction Signal in Parameter out

[ ]

n x

1 12 11

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

m

c c c

2 22 21

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

m

c c c

2 1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

Lm L L

c c c

e.g.:

1. Spectrum Estimation
2. Parameters for Recognition

Amplify or attenuate some frequency components of [ ]

n x

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Sinusoid Signals

– : amplitude (振幅) – : angular frequency (角頻率), – : phase (相角)

[ ]

( )

φ ω + = n A n x cos

ω φ A

T f π π ω 2 2 = =

[ ]

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 2 cos π ωn A n x

samples 25 = T

1 frequency normalized : ≤ ≤ f f

Period, represented by number of samples

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Sinusoid Signals (cont.)

is periodic with a period of N (samples)
Examples (sinusoid signals)

– is periodic with period N=8 – is periodic with period N=16 – is not periodic

[ ]

n x

[ ] [ ]

n x N n x = +

( ) ( )

φ ω φ ω + = + + n A N n A cos ) ( cos π ω 2 = N N π ω 2 =

[ ]

( )

4 / cos

1

n n x π =

[ ] ( )

8 / 3 cos

2

n n x π =

[ ] ( )

n n x cos

3

=

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Sinusoid Signals (cont.)

[ ] ( ) ( )

8 integers positive are and 8 2 4 4 4 cos ) ( 4 cos 4 cos 4 / cos

1 1 1 1 1 1

= ∴ ⋅ ⇒ ⋅ = ⇒ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = N k N k k N N n N n n n n x π π π π π π π

[ ] ( ) ( ) ( )

16 numbers positive are and 3 16 2 8 3 8 3 8 3 cos 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 n x π π π π π π π

[ ] ( ) ( ) ( ) ( ) ( )

! exist t doesn' integers positive are and 2 cos 1 cos 1 cos cos

3 3 3 3 3 3

N k N k N N n N n n n n x ∴ ⋅ = ⇒ + = + ⋅ = ⋅ = = Q π

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Sinusoid Signals (cont.)

Complex Exponential Signal

– Use Euler’s relation to express complex numbers

( )

φ φ

φ

sin cos j A Ae z jy x z

j

+ = = ⇒ + =

( )

number real a is A

Re Im

φ φ sin cos A y A x = =

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Sinusoid Signals (cont.)

A Sinusoid Signal

[ ]

( )

{ } { }

φ ω φ ω

φ ω

j n j n j

e Ae Ae n A n x Re Re cos = = + =

+

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Sinusoid Signals (cont.)

Sum of two complex exponential signals with

same frequency

– When only the real part is considered – The sum of N sinusoids of the same frequency is another sinusoid of the same frequency

( ) ( )

( )

φ ω φ ω φ φ ω φ ω φ ω + + +

= = + = +

n j j n j j j n j n j n j

Ae Ae e e A e A e e A e A

1 1

( ) ( ) ( )

φ ω φ ω φ ω + = + + + n A n A n A cos cos cos

1 1

numbers real are and ,

1

A A A

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Some Digital Signals

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Some Digital Signals

Any signal sequence can be represented

as a sum of shift and scaled unit impulse sequences (signals)

[ ]

n x

[ ] [ ] [ ]

k n k x n x

k

− =

∑

∞ −∞ =

δ

scale/weighted Time-shifted unit impulse sequence [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]

3 1 2 1 1 1 2 1 2 2 1 3 3 2 2 1 1 1 1 2 2

3 2

− + − − + − + + + − + + = − + − + − + + + − + + − = − = − =

∑ ∑

− = ∞ −∞ =

n n n n n n n x n x n x n x n x n x k n k x k n k x n x

k k

δ δ δ δ δ δ δ δ δ δ δ δ δ δ

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Digital Systems

A digital system T is a system that, given an

input signal x[n], generates an output signal y[n]

[ ] [ ]

{ }

n x T n y =

[ ]

n x

{ }

T

[ ]

n y

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Properties of Digital Systems

Linear

– Linear combination of inputs maps to linear combination of outputs

Time-invariant (Time-shift)

– A time shift of in the input by m samples give a shift in the output by m samples

[ ] [ ] { } [ ] { } [ ] { }

n x bT n x aT n bx n ax T

2 1 2 1

+ = +

[ ] [ ]

{ }

m m n x T m n y ∀ ± = ± ,

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Properties of Digital Systems (cont.)

Linear time-invariant (LTI)

– The system output can be expressed as a convolution (迴旋積分) of the input x[n] and the impulse response h[n] – The system can be characterized by the system’s impulse response h[n], which also is a signal sequence

If the input x[n] is impulse , the output is h[n]

[ ]

n δ

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Properties of Digital Systems (cont.)

Linear time-invariant (LTI)

– Explanation:

[ ] [ ] [ ]

k n k x n x

k

− =

∑

∞ −∞ =

δ

[ ]

{ }

[ ] [ ]

{ }

[ ] [ ]

{ }

[ ] [ ] [ ] [ ]

n h n x k n h k x k n T k x k n k x T n x T

k k k

∗ = − = − = − = ⇒

∑ ∑ ∑

∞ −∞ = ∞ −∞ = ∞ −∞ =

δ δ

scale Time-shifted unit impulse sequence Time invariant

Digital System

[ ]

n δ

[ ]

n h linear Time-invariant

[ ] [ ] [ ] [ ]

k n h k n n h n

T T

− ⎯→ ⎯ − ⎯→ ⎯ δ δ

Impulse response convolution

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Properties of Digital Systems (cont.)

Linear time-invariant (LTI)

– Convolution

Example

[ ]

n δ

0 1 2 1 3

2

[ ]

n h LTI

[ ]

n x

?

0 1 2 2 3 1 3 0 1 2 3 9

6

1 2 1 2 3 2 6

4

LTI

2 1 2 3 4 1 3

2

Length=M=3 Length=L=3 Length=L+M-1

[ ]

n δ ⋅ 3

[ ]

1 2 − ⋅ n δ

[ ]

2 1 − ⋅ n δ

3 1 11 2 1 3

1

4

2

Sum up

[ ]

n y

[ ]

n h ⋅ 3

[ ]

1 2 − ⋅ n h

[ ]

2 − n h

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Properties of Digital Systems (cont.)

Linear time-invariant (LTI)

– Convolution: Generalization

Reflect h[k] about the origin (→ h[-k])
Slide (h[-k] → h[-k+n] or h[-(k-n)] ), multiply it with

x[k]

Sum up

[ ]

k x

[ ]

k h

[ ]

k h −

Reflect Multiply slide Sum up

n

[ ] [ ] [ ] [ ] ( ) [ ]

n k h k x k n h k x n y

k k

− − = − =

∑ ∑

∞ −∞ = ∞ −∞ =

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0 1 2 1 3

2
1
2

1 3

2

0 1 2 2 3 1 3 1

1

1 3

2

1 11 2 1 1 3

2

1 2 3 2 1 1 3

2
1

3 4 3 2 1 3

2
2

4 3 1 11 2 1 3

1

4

2

Sum up

[ ]

k h

[ ]

k x

[ ]

k h −

[ ]

, = n n y

[ ]

1 + −k h

[ ]

2 + −k h

[ ]

3 + −k h

[ ]

4 + −k h

[ ]

1 , = n n y

[ ]

2 , = n n y

[ ]

3 , = n n y

[ ]

4 , = n n y

[ ]

n y

Reflect

[ ] [ ] [ ] [ ] [ ]

k n h k x n h n x n y

k

− = ∗ =

∑

∞ −∞ =

Convolution

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Properties of Digital Systems (cont.)

Linear time-invariant (LTI)

– Convolution is commutative and distributive

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

k n x k h k n h k x n x n h n h n x n y

k k

− = − = = =

∑ ∑

∞ −∞ = ∞ −∞ =

* *

[ ]

n h 1

[ ]

n h 2

[ ]

n h 2

[ ]

n h 1

[ ]

n h 2

[ ]

n h 1

[ ] [ ]

n h n h

2 1

+

– An impulse response has finite duration » Finite-Impulse Response (FIR) – An impulse response has infinite duration » Infinite-Impulse Response (IIR)

[ ] [ ] [ ] [ ] [ ] [ ]

n h n h n x n h n h n x

1 2 2 1

* * * * =

[ ] [ ] [ ] ( ) [ ] [ ] [ ] [ ]

n h n x n h n x n h n h n x

2 1 2 1

* * * + = +

Commutation Distribution

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Properties of Digital Systems (cont.)

Bounded Input and Bounded Output (BIBO): stable

– A LTI system is BIBO if only if h[n] is absolutely summable

[ ]

∞ ≤

∑

∞ −∞ = k

k h

[ ] [ ]

n B n y n B n x

y x

∀ ∞ < ≤ ∀ ∞ < ≤

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Properties of Digital Systems (cont.)

Causality

– A system is “casual” if for every choice of n0, the output sequence value at indexing n=n0 depends on only the input sequence value for n≤n0 – Any noncausal FIR can be made causal by adding sufficient long delay

[ ] [ ] [ ]

∑ ∑

= =

− + − =

K k M m k k k

m n x k n y n y

1

β α

z-1 z-1 z-1

M

β

2

β β

[ ]

n y

[ ]

n x

1

β

z-1 z-1 z-1

1

α

2

α

N

α

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Discrete-Time Fourier Transform (DTFT)

Frequency Response

– Defined as the discrete-time Fourier Transform of

–

is continuous and is periodic with period=

–

is a complex function of

[ ]

n h

( )

ω j

e H

π 2

( )

ω j

e H

ω

( ) ( ) ( ) ( )

( )

ω

ω ω ω ω

j

e H j j j i j r j

e e H e jH e H e H

∠

= + =

magnitude phase

( )

ω j

e H

proportional to two times of the sampling frequency

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Discrete-Time Fourier Transform (cont.)

Representation of Sequences by Fourier Transform

– A sufficient condition for the existence of Fourier transform

[ ]

∞ <

∑

∞ −∞ = n

n h [ ]

( ) ( ) [ ]

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 n 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

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Discrete-Time Fourier Transform (cont.)

Convolution Property

( )

[ ] [ ] [ ] [ ]

( )

[ ] [ ] [ ] [ ]

( ) ( )

[ ]

( ) ( )

ω ω ω ω ω ω ω ω ω ω j j j j n j n k k j n j n k j k n n j j

e H e X n h n x e H e X e n h e k x e k n h k x e Y k n h k x n h n x n y e n h e H ⇔ ∗ ∴ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − = − = ∗ = =

− ∞ −∞ = ∞ −∞ = − − ∞ −∞ = ∞ −∞ = ∞ −∞ = ∞ −∞ = −

∑ ∑ ∑ ∑ ∑ ∑

] [ ' ] [ ] [

' '

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

ω ω ω ω ω ω ω ω ω j j j j j j j j j

e H e X e Y e H e X e Y e H e X e Y ∠ + ∠ = ∠ ⇒ = ⇒ =

k n n k n n k n n − − = − ⇒ + = ⇒ − = ' ' '

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Discrete-Time Fourier Transform (cont.)

Parseval’s Theorem

– Define the autocorrelation of signal

[ ]

( )

∫ ∑

− ∞ −∞ =

=

π π ω

ω π d e X n x

j n 2 2

2 1

power spectrum

[ ]

n x

[ ]

( )

] [ ] [ ] [ ] [ ] [ ] [

* *

n x n x l n x l x m x n m x n R

l m xx

− ∗ = − − = + =

∗ ∞ −∞ = ∞ −∞ =

∑ ∑

( ) ( ) ( ) ( )

2 *

ω ω ω ω X X X S xx = = ⇔

[ ] ( ) ( )

∫ ∫

− −

= =

π π ω π π ω

ω ω π ω ω π d e X d e S n R

n j n j xx xx 2

2 1 2 1

[ ] [ ] [ ] [ ] ( )

∑ ∫ ∑

∞ −∞ = − ∞ −∞ =

= = = =

m m xx

d X m x m x m x R n

π π

ω ω π

2 2 *

2 1 Set

The total energy of a signal can be given in either the time or frequency domain.

) ( l n n l m n m l − − = − = ⇒ + =

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Discrete-Time Fourier Transform (cont.)

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Z-Transform

z-transform is a generalization of (Discrete-Time) Fourier

transform

– z-transform of is defined as

Where , a complex-variable
For Fourier transform

– z-transform evaluated on the unit circle

( )

[ ]

∑

∞ −∞ = −

=

n n

z n h z H

[ ]

( )

z H n h

[ ]

( )

ω j

e H n h

[ ]

n h

ω j

re z =

( )

ω

j

e z j

z H e H

=

) 1 ( = = z e z

jω

complex plan

unit circle Im Re

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32

Z-Transform (cont.)

Fourier transform vs. z-transform

– Fourier transform used to plot the frequency response of a filter – z-transform used to analyze more general filter characteristics, e.g. stability

ROC (Region of Converge)

– Is the set of z for which z-transform exists (converges) – In general, ROC is a ring-shaped region and the Fourier transform exists if ROC includes the unit circle

[ ]

∞ <

∑

∞ −∞ = − n n

z n h

complex plan

R1 R2 Re Im

absolutely summable

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Z-Transform (cont.)

An LTI system is defined to be causal, if its impulse

response is a causal signal, i.e.

– Similarly, anti-causal can be defined as

An LTI system is defined to be stable, if for every

bounded input it produces a bounded output

– Necessary condition:

That is Fourier transform exists, and therefore z-transform

include the unit circle in its region of converge

[ ]

for < = n n h

[ ]

for > = n n h

Right-sided sequence Left-sided sequence

[ ]

∞ <

∑

∞ −∞ = n

n h

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

k n x k h k n h k x n x n h n h n x n y

k k

− = − = = =

∑ ∑

∞ −∞ = ∞ −∞ =

* *

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Z-Transform (cont.)

Right-Sided Sequence

– E.g., the exponential signal

[ ] [ ] [ ]

⎩ ⎨ ⎧ < ≥ = = for for 1 ere wh , . 1

1

n n n u n u a n h

n

( )

1 1 1

1 1

− ∞ −∞ = − − ∞ −∞ =

− = = =

∑ ∑

az az z a z H

n n n n n

If

1

1 < −

az

a z ROC > ∴ is

1

Re Im

a

[ ]

1 if exists

f

ansform Fourier tr

1

< a n h

×

have a pole at (Pole: z-transform goes to infinity)

a z =

the unit cycle

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Z-Transform (cont.)

Left-Sided Sequence

– E.g.

[ ] [ ]

1 . 2

2

− − − = n u a n h

n

( ) [ ]

( )

1 1 1 1 1 1 1 2

1 1 1 1 1 1 1 1

− − − − ∞ = − ∞ = − − − −∞ = − ∞ −∞ =

− = − − − = − − = − = − = − = − − − =

∑ ∑ ∑ ∑

az z a z a z a z a z a z a z n u a z H

n n n n n n n n n n n

If

1

<

− z

a

a z ROC < ∴ is

2

Re Im

a

[ ] [ ]

−∞ → < n n h n h a as lly exponentia go will because exist, t doesn'

f

ansform Fourier tr the , 1 when

2 2

×

the unit cycle

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Z-Transform (cont.)

Two-Sided Sequence

– E.g.

[ ] [ ] [ ]

1 2 1 3 1 . 3

3

− − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− = n u n u n h

n n

[ ] [ ]

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'

f

ansform Fourier tr

3 3

ROC n h

3 1 and 2 1 is

3

> < ∴ z z ROC

( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − + + =

− −

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

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Z-Transform (cont.)

Finite-length Sequence

– E.g.

[ ]

⎩ ⎨ ⎧ − ≤ ≤ =

thers

, 1 , . 3

4

N n a n h

n

( )

( ) ( )

a z a z z az az az z a z H

N N N N N n n N n n n

− − = − − = = =

− − − − = − − = −

∑ ∑

1 1 1 1 1 1 4

1 1 1

except plane

entire

is

4

= ∴ z z ROC

1 3 2 2 1

.....

− − − −

+ + + +

N N N N

a z a az z Im

×

the unit cycle

3 1 −

7 poles at zero A pole and zero at is cancelled

a z =

4 π

( )

1 1 ,

2

− = = ,..,N k ae z

N k j k π

If N=8

N-1

Re

SLIDE 38

2004 Speech - Berlin Chen

38

Z-Transform (cont.)

Properties of z-transform
1. If is right-sided sequence, i.e. and if ROC

is the exterior of some circle, the all finite for which will be in ROC

If ,ROC will include
2. If is left-sided sequence, i.e. , the ROC is

the interior of some circle,

If ,ROC will include
3. If is two-sided sequence, the ROC is a ring
4. The ROC can’t contain any poles

[ ]

n h

[ ]

1

, n n n h ≤ =

1 ≥

n r z > z

∞ = z [ ]

n h

[ ]

2

, n n n h ≥ =

2 <

n = z

[ ]

n h

A causal sequence is right-sided with ROC is the exterior of circle including

1 ≥

n ∞ = z

∴

SLIDE 39

2004 Speech - Berlin Chen

39

Summary of the Fourier and z-transforms

SLIDE 40

2004 Speech - Berlin Chen

40

LTI Systems in the Frequency Domain

Example 1: A complex exponential

sequence

– System impulse response – Therefore, a complex exponential input to an LTI system results in the same complex exponential at the output, but modified by

The complex exponential is an eigenfunction of an LTI

system, and is the associated eigenvalue

[ ]

n h

[ ] [ ] [ ] [ ] [ ]

( )

n j j k j n j k n j

e e H e k h e e k h n h n x n y

ω ω ω ω ω

= = = ∗ =

− ∞ ∞ = − ∞ ∞ =

∑ ∑

k

) (

k

( )

response. frequency system the as to referred

ften

is It response. impulse system the

f

ansform Fourier tr the :

ω j

e H

[ ]

n j

e n x

ω

=

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

k n x k h k n h k x n x n h n h n x n y

k k

− = − = = =

∑ ∑

∞ −∞ = ∞ −∞ =

* *

[ ] { }

( ) [ ]

n x e H n x T

jω

=

( )

jω

e H

( )

jω

e H

SLIDE 41

2004 Speech - Berlin Chen

41

[ ]

( ) ( ) ( ) ( )

[ ]

( )

[ ]

( ) ( )

[ ]

) ( ) ( ) ( ) (

cos 2 2 2 2

jω jω n j e H j jω n j e H j jω n j jω * n j jω n j j jω n j j jω

e H n e H A e e e H e e e H A e e H e e H A e e A e H e e A e H n y

jω jω

∠ + + = + = + = + =

+ − ∠ − + ∠ + − + − − −

φ ω

φ ω φ ω φ ω φ ω ω φ ω φ

LTI Systems in the Frequency Domain (cont.)

Example 2: A sinusoidal sequence

– System impulse response

[ ] ( )

φ + = n w A n x cos

[ ] ( )

n j j n j j

e e A e e A n A n x 2 2 cos

ω φ ω φ

φ ω

− −

+ = + =

[ ]

n h

( )

θ θ θ θ

θ θ θ θ θ

j j j j

e e i e i e

− −

+ = ⇒ − = + = 2 1 cos sin cos sin cos

( ) ( ) ( ) ( )

( )

jω

e H j jω jω * jω * jω

e e H e H e H e H

∠ − −

= =

( )

( ) ( )

ω ω ω ω ω ω ω ω

ω ω ω

sin cos sin cos sin cos * sin cos

*

j j e j e jy x z j e jy x z

j j j

− = − + − = − = ⇒ − = + = ⇒ + =

−

SLIDE 42

2004 Speech - Berlin Chen

42

LTI Systems in the Frequency Domain (cont.)

Example 3: A sum of sinusoidal sequences

– A similar expression is obtained for an input consisting of a sum

f complex exponentials

[ ]

( )

∑

=

+ =

K k k k k

n A n x

1

cos φ ω

[ ]

( ) ( )

[ ]

∑

=

∠ + + =

K k j k k j k

k k

e H n e H A n y

1

cos

ω ω

φ ω

SLIDE 43

2004 Speech - Berlin Chen

43

LTI Systems in the Frequency Domain (cont.)

Example 4: Convolution Theorem

[ ] [ ]

∑

∞ −∞ =

− =

k

kP n n x δ

[ ] [ ]

1 , < = ∑

∞ −∞ =

a n u a n h

k n

( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ∑

∞ −∞ =

k P P e X

k j

π ω δ π

ω

2 2

( )

ω ω j j

ae e H

−

− = 1 1

DTFT DTFT

( ) ( ) ( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = =

∑ ∑

∞ −∞ = − ∞ −∞ = −

k P ae P k P P ae e X e H e Y

k k P j k j j j j

π ω δ π π ω δ π

π ω ω ω ω

2 1 1 2 2 2 1 1

2

[ ] [ ]

( ) ( )

ω ω j j

e H e X n h n x ⇔ ∗

SLIDE 44

2004 Speech - Berlin Chen

44

LTI Systems in the Frequency Domain (cont.)

Example 5: Windowing Theorem

[ ] [ ]

( ) ( )

ω ω

π

j j

e X e W n w n x ∗ ⇔ 2 1

[ ] [ ]

∑

∞ −∞ =

− =

k

kP n n x δ [ ]

⎪ ⎩ ⎪ ⎨ ⎧ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − =

therwise

1 ,......, 1 , , 1 2 cos 46 . 54 . N n N n n w π

Hamming window

( ) ( ) ( ) ( ) ( ) ( )

∑ ∑ ∑ ∑ ∑

∞ −∞ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ∞ −∞ = ∞ = −∞ = ∞ −∞ = ∞ −∞ =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ∗ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ∗ = ∗ =

k k P j k m m jm k j k j j j j

e W P m k P e W P k P e W P k P P e W e X e W e Y

π ω ω ω ω ω ω

π ω δ π ω δ π ω δ π π π

2

1 2 1 2 1 2 2 2 1 2 1

SLIDE 45

2004 Speech - Berlin Chen

45

Difference Equation Realization for a Digital Filter

The relation between the output and input of a digital

filter can be expressed by

[ ] [ ] [ ]

∑ ∑

= =

− + − =

N k M k k k

k n x k n y n y

1 0 β

α

( ) ( ) ( )

k N k M k k k k

z z X z z Y z Y

− = = −

∑ ∑

+ =

1 0 β

α

delay property

( ) ( ) ( )

∑ ∑

= − = −

− = =

N k k k M k k k

z z z X z Y z H

1

1 α β

[ ] ( ) [ ] ( )

n

z z X n n x z X n x

−

→ − →

linearity and delay properties A rational transfer function Causal: Rightsided, the ROC outside the

utmost pole

Stable: The ROC includes the unit circle Causal and Stable: all poles must fall inside the unit circle (not including zeros)

z-1 z-1 z-1

M

β

2

β β

[ ]

n y

[ ]

n x

1

β

z-1 z-1 z-1

1

α

2

α

N

α

SLIDE 46

2004 Speech - Berlin Chen

46

Difference Equation Realization for a Digital Filter (cont.)

SLIDE 47

2004 Speech - Berlin Chen

47

Magnitude-Phase Relationship

Minimum phase system:

– The z-transform of a system impulse response sequence ( a rational transfer function) has all zeros as well as poles inside the unit cycle – Poles and zeros called “minimum phase components” – Maximum phase: all zeros (or poles) outside the unit cycle

All-pass system:

– Consist a cascade of factor of the form – Characterized by a frequency response with unit (or flat) magnitude for all frequencies

1 1

± − ⎥

⎦ ⎤ ⎢ ⎣ ⎡ − az z

a*

Poles and zeros occur at conjugate reciprocal locations 1 1 1

1 =

−

az z

a*

SLIDE 48

2004 Speech - Berlin Chen

48

Magnitude-Phase Relationship (cont.)

Any digital filter can be represented by the cascade of a

minimum-phase system and an all-pass system

( ) ( ) ( )

z H z H z H

ap min

=

( ) ( ) ( ) ( )(

)

( ) ( )(

) ( ) ( )

( )(

) ( ) ( )

filter. pass

all

a is 1 1 filter. phase minimum a also is 1 : where 1 1 1 filter) phase minimum a is ( 1 : as expressed be can circle. unit the

utside

1) ( 1 zero

ne
nly

has that Suppose

1 * 1 1 1 * 1 1 1 * 1 − − − −

− − − − − − = − = < az z a az z H az z a az z H z H z a z H z H z H a a* z H

SLIDE 49

2004 Speech - Berlin Chen

49

FIR Filters

FIR (Finite Impulse Response)

– The impulse response of an FIR filter has finite duration – Have no denominator in the rational function

No feedback in the difference equation

– Can be implemented with simple a train of delay, multiple, and add operations

( )

z H

[ ] [ ]

∑

=

− =

M r r

r n x n y

0 β

z-1 z-1 z-1

M

β

1

β β

[ ]

n y

[ ]

n x

[ ]

⎩ ⎨ ⎧ ≤ ≤ =

therwise

, , M n n h

n

β

( ) ( ) ( )

∑

= −

= =

M k k k z

z X z Y z H

0 β

SLIDE 50

2004 Speech - Berlin Chen

50

First-Order FIR Filters

A special case of FIR filters

[ ] [ ] [ ]

1 − + = n x n x n y α

( )

1

−

+ = z z H α

( )

ω ω

α

j j

e e H

−

+ = 1

( )

( ) ( ) ( )

( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = + = + + = − + = 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

SLIDE 51

2004 Speech - Berlin Chen

51

Discrete Fourier Transform (DFT)

The Fourier transform of a discrete-time sequence is a

continuous function of frequency

– We need to sample the Fourier transform finely enough to be able to recover the sequence – For a sequence of finite length N, sampling yields the new transform referred to as discrete Fourier transform (DFT)

( )

[ ] [ ] [ ]

1 , 1 1 ,

2 1 2 1

− ≤ ≤ = − ≤ ≤ =

∑ ∑

− = − − =

N n e k X N n x N n e n x k X

kn N j N k kn N j N n π π

Inverse DFT, Synthesis DFT, Analysis

SLIDE 52

2004 Speech - Berlin Chen

52

Discrete Fourier Transform (cont.)

[ ] [ ]

( ) ( ) ( ) ( )

[ ] [ ] [ ] [ ] [ ] [ ]⎥

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ≤ ≤ = − ≤ ≤ ∀

− ⋅ − − ⋅ − − − ⋅ − ⋅ − − − =

∑

1 1 1 1 1 1 1 1 1 1 , 1

1 1 2 1 1 2 1 1 2 1 1 2 2 1

N X X X N x x x e e e e N n e n x k X N k

N N N j N N j N N j N j kn N j N n

M M L M L M M L L

π π π π π

SLIDE 53

2004 Speech - Berlin Chen

53

Discrete Fourier Transform (cont.)

Orthogonality of Complex Exponentials

( )

⎩ ⎨ ⎧ = =

∑

− =

−

therwise

, if , 1 1

1

2

mN k-r e N

N n

n r k N j π

[ ] [ ] [ ] [ ]

( )

[ ]

( )

[ ] [ ] [ ]

kn N j n r k N j n r k N j rn N j kn N j

e n x k X r X e N k X e k X N e n x e k X N n x

N r N n N k N n N k N n N k

π π π π π 2 2 2 2 2

1 1 1 1 1 1 1

1 1 1

− − − −

∑ ∑ ∑ ∑ ∑ ∑ ∑

− = − = − = − = − = − = − =

= ⇒ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = = ⇒ = [ ] [ ] [ ]

r X mN r X k X = + =

SLIDE 54

2004 Speech - Berlin Chen

54

Discrete Fourier Transform (DFT)

Parseval’s theorem

[ ]

( )

∑ ∑

− = − =

=

1 2 1 2

1

N k N n

k X N n x

Energy density