[PPT] - Speech Signal Representations Berlin Chen 2003 References: 1. X. PowerPoint Presentation

SLIDE 1

Speech Signal Representations

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 2003

SLIDE 2

2

Introduction

Current speech recognition systems are mainly

composed of :

– A front-end feature extractor (feature extraction module)

Required to discover salient characteristics suited for classification
Based on scientific and/or heuristic knowledge about patterns to

recognize

– A back-end classifier (classification module)

Required to set class boundaries accurately in the feature space
Statistically designed according to the fundamental Bayes’ decision

theory

SLIDE 3

3

Background Review: Background Review:

Digital Signal Processing

SLIDE 4

4

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

SLIDE 5

5

Analog Signal to Digital Signal

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

SLIDE 6

6

Analog Signal to Digital Signal

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

SLIDE 7

7

Analog Signal to Digital Signal

( )

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

SLIDE 8

8

Analog Signal to Digital Signal

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

SLIDE 9

9

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

SLIDE 10

10

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

SLIDE 11

11

Sinusoid Signals

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

=

SLIDE 12

12

Sinusoid Signals

[ ] ( ) ( )

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

SLIDE 13

13

Sinusoid Signals

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 = =

SLIDE 14

14

Sinusoid Signals

A Sinusoid Signal

[ ]

( )

{ } { }

φ ω φ ω

φ ω

j n j n j

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

+

SLIDE 15

15

Sinusoid Signals

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

SLIDE 16

16

Some Digital Signals

SLIDE 17

17

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

SLIDE 18

18

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

SLIDE 19

19

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 ∀ ± = ± ,

SLIDE 20

20

Properties of Digital Systems

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

SLIDE 21

21

Properties of Digital Systems

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

SLIDE 22

22

Properties of Digital Systems

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

SLIDE 23

23

Properties of Digital Systems

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

− − = − =

∑ ∑

∞ −∞ = ∞ −∞ =

SLIDE 24

24 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

SLIDE 25

25

Properties of Digital Systems

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

SLIDE 26

26

Properties of Digital Systems

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

∀ ∞ < ≤ ∀ ∞ < ≤

SLIDE 27

27

Properties of Digital Systems

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

SLIDE 28

28

Discrete-Time Fourier Transform (DTFT)

Frequency Response

– Defined as the discrete-time Fourier Transform

f

– 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

SLIDE 29

29

Discrete-Time Fourier Transform

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

SLIDE 30

30

Discrete-Time Fourier Transform

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 − − = − ⇒ + = ⇒ − = ' ' '

SLIDE 31

31

Discrete-Time Fourier Transform

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 − − = − = ⇒ + =

SLIDE 32

32

Discrete-Time Fourier Transform

SLIDE 33

33

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

SLIDE 34

34

Z-Transform

Fourier transform vs. z-transform

– Fourier transform used to plot the frequency response

f 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

SLIDE 35

35

Z-Transform

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

− = − = = =

∑ ∑

∞ −∞ = ∞ −∞ =

* *

SLIDE 36

36

Z-Transform

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

SLIDE 37

37

Z-Transform

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

SLIDE 38

38

Z-Transform

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

SLIDE 39

39

Z-Transform

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 40

40

Z-Transform

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 41

41

Summary of the Fourier and z-transforms

SLIDE 42

42

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 43

43

[ ]

( ) ( ) ( ) ( )

[ ]

( )

[ ]

( ) ( )

[ ]

) ( ) ( ) ( ) (

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

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 44

44

LTI Systems in the Frequency Domain

Example 3: A sum of sinusoidal sequences

– A similar expression is obtained for an input consisting of a sum of 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 45

45

LTI Systems in the Frequency Domain

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 46

46

LTI Systems in the Frequency Domain

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 47

47

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 48

48

Difference Equation Realization for a Digital Filter

SLIDE 49

49

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 50

50

Magnitude-Phase Relationship

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 51

51

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 52

52

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 53

53

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 54

54

Discrete Fourier Transform (DFT)

[ ] [ ]

( ) ( ) ( ) ( )

[ ] [ ] [ ] [ ] [ ] [ ]

           − =             −               − ≤ ≤ = − ≤ ≤ ∀

− ⋅ − − ⋅ − − − ⋅ − ⋅ − − − =

∑

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 55

55

Discrete Fourier Transform (DFT)

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 56

56

Discrete Fourier Transform (DFT)

Parseval’s theorem

[ ]

( )

∑ ∑

− = − =

=

1 2 1 2

1

N k N n

k X N n x

Energy density