sp2004f lecture07-01 digital signal processingberlin.csie.ntnu.edu.tw/pastcourses/2004-tcfst-audio...

Digital Signal Processing

References:1. X. Huang et. al., Spoken Language Processing, Chapters 5, 62. J. R. Deller et. al., Discrete-Time Processing of Speech Signals, Chapters 4-63. J. W. Picone, “Signal modeling techniques in speech recognition,”

proceedings of the IEEE, September 1993, pp. 1215-1247

Berlin Chen 2004

2004 Speech - Berlin Chen 2

Analog Signal to Digital Signal

[ ] ( ) period sampling : , TnTxnx a=

rate sampling 1TFs =

Analog Signal

Discrete-time Signal or Digital Signal

nTt =

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

Digital Signal:Discrete-time

signal with discreteamplitude


Analog Signal to Digital Signal (cont.)

Impulse TrainTo

Sequence [ ] ( ))( ˆ nTxnx a=

Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )∑∑

∑∞

−∞=

∞

−∞=

∞

−∞=

−=−=

−==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a

Continuous-TimeSignal

Discrete-TimeSignal

Digital Signal

Discrete-time signal with discrete

amplitude

[ ] nx

Continuous-Time to Discrete-Time Conversion

( ) ( )∑∞−∞=

−=n

nTtts δPeriodic Impulse Train ( ) [ ]nxtxs by specifieduniquely becan

switch

( )tx a

( ) ( )∑∞−∞=

−=n

nTtts δ

( )

( )∫∞

∞−

=

≠∀=

1

0 ,0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T



• A continuous signal sampled at different periods

T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )∑∑

∑∞

−∞=

∞

−∞=

∞

−∞=

−=−=

−==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa

frequency) (sampling22 ss FTππ ==Ω

⎟⎠⎞

⎜⎝⎛=ΩΩ

TsNπ

21

aliasing distortion

( ) ( )∑∞−∞=

Ω−Ω=Ωk

skTjS δπ2

( ) ( ) ( )

( ) ( )( )∑∞

−∞=

Ω−Ω=Ω

Ω∗Ω=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧

Ω>Ω⇒Ω>Ω−Ω

2 NsNNsQ

( )⎭⎬⎫

⎩⎨⎧

Ω



• 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 sΩ

( ) tth sΩ= sinc( ) ( ) ( )

( ) ( )∑

∑∞

−∞=

∞

−∞=

−Ω=

−=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns Ω>Ω 2


Two Main Approaches to Digital Signal Processing

• Filtering

• Parameter Extraction

FilterSignal in Signal out

[ ] nx [ ] ny

ParameterExtraction

Signal in Parameter out

[ ] nx

1

12

11

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

mc

cc

2

22

21

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

mc

cc

2

1

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

Lm

L

L

c

cc

e.g.:1. Spectrum Estimation2. Parameters for Recognition

Amplify or attenuate some frequencycomponents of [ ] nx


Sinusoid Signals

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

[ ] ( )φω += nAnx cos

ωφ

A

Tf ππω 22 ==

[ ] ⎟⎠⎞

⎜⎝⎛ −=

2cos πωnAnx

samples 25=T

10frequency normalized:

≤≤ ff

Period, represented by number of samples


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

[ ]nx[ ] [ ]nxNnx =+

( ) ( )φωφω +=++ nANnA cos)(cosπω 2=N

Nπω 2=

[ ] ( )4/cos1 nnx π=[ ] ( )8/3cos2 nnx π=[ ] ( )nnx cos3 =


Sinusoid Signals (cont.)[ ] ( )

( )

8

integers positive are and 824

44cos)(

4cos

4cos

4/cos

1

11

11

1

=∴

⋅⇒⋅=⇒

⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛=

=

N

kNkkN

NnNnn

nnx

ππ

πππππ

[ ] ( )

( )

( )

16

numbers positive are and 3

1628

3

83

83cos

83cos

83cos

8/3cos

2

222

22

2

=∴

=⇒⋅=⋅⇒

⎟⎠⎞

⎜⎝⎛ ⋅+⋅=⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ ⋅=

=

N

kNkNkN

NnNnn

nnx

ππ

πππππ

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

!exist t doesn' integers positive are and

2 cos1cos1cos

cos

3

3

3

33

3

NkN

kNNnNnn

nnx

∴

⋅=⇒+=+⋅=⋅=

=

Q

π



• Complex Exponential Signal– Use Euler’s relation to express complex numbers

( )φφφ sincos

jAAez

jyxzj +==⇒

+=( )number real a is A

Re

Im

φφ

sin cos

AyAx

==



• A Sinusoid Signal

[ ] ( )( ){ }

{ }φωφω

φω

jnj

nj

eAeAe

nAnx

Re Re

cos

=

=

+=+



• 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

( ) ( )

( )

( )φω

φω

φφω

φωφω

+

++

=

=

+=

+

nj

jnj

jjnj

njnj

Ae

Aee

eAeAe

eAeA10

10

10

10

( ) ( ) ( )φωφωφω +=+++ nAnAnA coscoscos 1100

numbers real are and , 10 AAA


Some Digital Signals


Some Digital Signals

• Any signal sequence can be represented as a sum of shift and scaled unit impulse sequences (signals)

[ ]nx

[ ] [ ] [ ]knkxnxk

−= ∑∞

−∞=δ

scale/weightedTime-shifted unit impulse sequence

[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2

−+−−+−+++−++=−+−+−+++−++−=

−=−= ∑∑−=

∞

−∞=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk

δδδδδδδδδδδδ

δδ


Digital Systems

• A digital system T is a system that, given an input signal x[n], generates an output signal y[n]

[ ] [ ]{ }nxTny =

[ ]nx { } T [ ]ny


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

[ ] [ ]{ } [ ]{ } [ ]{ }nxbTnxaTnbxnaxT 2121 +=+

[ ] [ ]{ } mmnxTmny ∀±=± ,


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][ ]nδ



• Linear time-invariant (LTI)– Explanation:

[ ] [ ] [ ]knkxnxk

−= ∑∞

−∞=δ

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

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

∗=

−=

−=

−=⇒

∑

∑

∑

∞

−∞=

∞

−∞=

∞

−∞=

δ

δ

scale

Time-shifted unit impulse sequence

Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T

−⎯→⎯−

⎯→⎯

δ

δ

Impulse response

convolution



• Linear time-invariant (LTI)– Convolution

• Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx ?

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδ⋅3

[ ]12 −⋅ nδ

[ ]21 −⋅ nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nh⋅3

[ ]12 −⋅ nh

[ ]2−nh



• 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 [ ]kx

[ ]kh

[ ]kh −

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

−−=

−=

∑

∑∞

−∞=

∞

−∞=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh −[ ] 0, =nny

[ ]1+−kh

[ ]2+−kh

[ ]3+−kh

[ ]4+−kh

[ ] 1, =nny

[ ] 2, =nny

[ ] 3, =nny

[ ] 4, =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

k−=

∗=

∑∞

−∞=

Convolution



• Linear time-invariant (LTI)– Convolution is commutative and distributive

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

−=

−=

==

∑

∑

∞

−∞=

∞

−∞=

* *

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +

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

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

[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

****

=

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

nhnhnx

21

21

***

+=+

Commutation

Distribution



• Bounded Input and Bounded Output (BIBO): stable

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

[ ] ∞≤∑∞−∞=k

kh

[ ][ ] nBny

nBnx

y

x

∀∞



• 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

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx1β

z-1

z-1

z-1

1α

2α

Nα


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

[ ]nh( )ωjeH

π2( )ωjeH

ω

( ) ( ) ( )( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeH∠=

+=

magnitudephase

( )ωjeH

proportional to twotimes of the samplingfrequency


Discrete-Time Fourier Transform (cont.)

• Representation of Sequences by Fourier Transform

– A sufficient condition for the existence of Fourier transform

[ ] ∞


Discrete-Time Fourier Transform (cont.)• Convolution Property

( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

⇔∗∴

=

⎟⎠

⎞⎜⎝

⎛=

−=

−=∗=

=

−∞

−∞=

∞

−∞=

−

−∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

−

∑∑

∑ ∑

∑

∑

][

'

][][

'

'

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

∠+∠=∠⇒

=⇒

=

knnknnknn

−−=−⇒+=⇒−=

''

'



• Parseval’s Theorem

– Define the autocorrelation of signal

[ ] ( )∫∑ −∞−∞=

= ππω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

*

*

nxnxlnxlx

mxnmxnR

l

mxx

−∗=−−=

+=

∗∞

−∞=

∞

−∞=

∑

∑

( ) ( ) ( ) ( ) 2* ωωωω XXXS xx ==⇔

[ ] ( ) ( )∫∫ −− ==π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx2

21

21

[ ] [ ] [ ] [ ] ( )∑ ∫∑∞

−∞=−

∞

−∞=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22*

210

0Set

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

)(

lnnlmnml

−−=−=⇒+=


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

nznhzH

[ ] ( )zHnh [ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω jezj zHeH ==)1( == zez jω

complex plan

unit circle

Im

Re


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

[ ] ∞


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

[ ] 0for 0 = nnh

Right-sided sequence

Left-sided sequence

[ ] ∞


Z-Transform (cont.)

• Right-Sided Sequence– E.g., the exponential signal

[ ] [ ] [ ]⎩⎨⎧

<≥

==0for 00for 1

ere wh, .1 1 nn

nunuanh n

( ) ( ) 111 11

−

∞

−∞=

−−∞

−∞= −=== ∑∑

azazzazH

n

nn

n

n

If 11 ∴ is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1


Z-Transform (cont.)

• Left-Sided Sequence– E.g.

[ ] [ ]1 .2 2 −−−= nuanh n( ) [ ]

( ) 111

10

1

1

12

11

11111

1

−−

−

−

∞

=

−∞

=

−

−−

−∞=

−∞

−∞=

−=

−−

−=−

−=−=−=

−=−−−=

∑∑

∑∑

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11


Z-Transform (cont.)

• Two-Sided Sequence– E.g.

[ ] [ ] [ ]121

31 .3 3 −−⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−= nununh

nn

[ ]

[ ]21 ,

211

1 121

31 ,

311

1 31

1

1

<−

⎯→←−−⎟⎠⎞

⎜⎝⎛−

>+

⎯→←⎟⎠⎞

⎜⎝⎛ −

−

−

zz

nu

zz

nu

zn

zn

Re

Im

×

the unit cycle

×21

31

− [ ]circleunit theincludet doesn' because

exist,t doesn' of ansformFourier tr

3

3

ROCnh

31 and

21 is 3 >


Z-Transform (cont.)

• Finite-length Sequence– E.g.

[ ]⎩⎨⎧ −≤≤

=others ,0

10 , .3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

−−

=−

−=== −−

−−

=

−−

=

− ∑∑ 11

11

0

11

04

11

1

0except plane- entire is 4 =∴ zzROC

13221 ..... −−−− ++++ NNNN azaazz

Im

×

the unit cycle

31

−

7 poles at zero A pole and zero at is cancelled az =

4π

( ) 11 ,2

−== ,..,N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont.)

• Properties of z-transform1. 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 ring4. The ROC can’t contain any poles

[ ]nh [ ] 1 ,0 nnnh ≤=

01 ≥n

0rz >z

∞=z

[ ]nh [ ] 2 ,0 nnnh ≥=

02


Summary of the Fourier and z-transforms


LTI Systems in the Frequency Domain

• Example 1: A complex exponential sequence– System impulse response

– Therefore, a complex exponential input to an LTI systemresults 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

[ ]nh[ ] [ ] [ ] [ ]

[ ]

( ) njjkjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=∗=

−∞

∞=

−∞

∞=

∑

∑

-k

)(

-k( )

response.frequency system theas toreferredoften isIt

response. impulse system the of ansformFourier tr the:ωjeH

[ ] njenx ω= [ ] [ ] [ ][ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

−=

−=

==

∑

∑

∞

−∞=

∞

−∞=

* *

[ ]{ } ( ) [ ]nxeHnxT jω=( )jωeH

( )jωeH


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjω*njjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

∠++=

+=

+=

+=

+−∠−+∠

+−+

−−−

φω

φωφω

φωφω

ωφωφ

LTI Systems in the Frequency Domain (cont.)

• Example 2: A sinusoidal sequence

– System impulse response

[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0ωφωφ

φω

−−+=

+=

[ ]nh( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

−

−

+=⇒

−=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )000

00

jωeHjjωjω*

jω*jω

eeHeH

eHeH∠−

−

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos *

sincos *

jje

jejyxz

jejyxz

j

j

j

−=−+−=

−=⇒−=

+=⇒+=

−



• Example 3: A sum of sinusoidal sequences

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

[ ] ( )∑=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]∑=

∠++=K

k

jkk

jk

kk eHneHAny1

cos ωω φω



• Example 4: Convolution Theorem

[ ] [ ]∑∞−∞=

−=k

kPnnx δ

[ ] [ ] 1 ,



• Example 5: Windowing Theorem [ ] [ ] ( ) ( )ωωπ

jj eXeWnwnx ∗⇔21

[ ] [ ]∑∞−∞=

−=k

kPnnx δ

[ ]⎪⎩

⎪⎨⎧

−=⎟⎠⎞

⎜⎝⎛

−−=

otherwise 0

1,......,1,0 ,1

2cos46.054.0 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

∑

∑ ∑

∑

∑

∞

−∞=

⎟⎠⎞

⎜⎝⎛ −

∞

−∞=

∞=

−∞=

∞

−∞=

∞

−∞=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ −−=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ −∗=

⎟⎠⎞

⎜⎝⎛ −∗=

∗=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


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

kkk knxknyny

1 0βα

( ) ( ) ( ) kNk

M

kk

kk zzXzzYzY

−

= =

−∑ ∑+=1 0

βα

delay property

( ) ( )( ) ∑

∑

=

−

=

−

−== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 00 nzzXnnx

zXnx−→−

→linearity and delay properties

A rational transfer functionCausal:Rightsided, the ROC outside the outmost poleStable:The ROC includes the unit circleCausal and Stable:all poles must fall inside the unit circle (not including zeros)

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx1β

z-1

z-1

z-1

1α

2α

Nα


Difference Equation Realization for a Digital Filter (cont.)


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 responsewith unit (or flat) magnitude for all frequencies

1

111

±

− ⎥⎦

⎤⎢⎣

⎡

− azz-a*

Poles and zeros occur at conjugate reciprocal locations

111

1 =− −azz-a*


Magnitude-Phase Relationship (cont.)

• Any digital filter can be represented by the cascade of a minimum-phase system and an all-pass system

( ) ( ) ( )zHzHzH apmin=

( )( )

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

( )( )( )( ) filter. pass-all a is 11

filter. phase minimum a also is 1

:where

111

filter) phase minimum a is ( 1

:as expressed becan circle.unit theoutside

1)( 1 zero oneonly has that Suppose

1

*

11

1

*1

1

1*

1

−

−

−−

−−

−

−−

−=

−=

<

azza

azzH

azzaazzH

zHzazHzH

zH

aa*zH


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

( )zH

[ ] [ ]∑=

−=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ ≤≤

=otherwise ,00 , Mn

nh nβ

( ) ( )( ) ∑=−==

M

k

kk zzX

zYzH0β


First-Order FIR Filters

• A special case of FIR filters

[ ] [ ] [ ]1−+= nxnxny α ( ) 11 −+= zzH α( ) ωω α jj eeH −+= 1( ) ( )

( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+−=

+=++=

−+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH


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)

( ) [ ]

[ ] [ ] 10 , 1

10 ,

21

0

21

0

−≤≤=

−≤≤=

∑

∑−

=

−−

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π

Inverse DFT, Synthesis

DFT, Analysis


Discrete Fourier Transform (cont.)

[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−≤≤=

−≤≤∀

−⋅−−⋅−−

−⋅−⋅−

−−

=∑

1

10

1

10

1

1

111

10 ,

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π


Discrete Fourier Transform (cont.)

• Orthogonality of Complex Exponentials ( )

⎩⎨⎧ =

=∑−

=

−

otherwise ,0 if ,11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

−

−

−−

∑

∑∑

∑ ∑∑

∑

−

=

−

=

−

=

−

=

−

=

−

=

−

=

=⇒

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=⇒

=

[ ] [ ][ ]rX

mNrXkX=

+=


Discrete Fourier Transform (DFT)

• Parseval’s theorem

[ ] ( )∑∑−

=

−

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density

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