Digital Signal Processing Introduction Course R.WEBER SP2 ESI module Traitement Numérique

Digital Signal Processing Introduction Course

R.WEBER

SP2 ESI module Traitement NumériqueA] Time and Frequency description of a digital signalB] Digital Filtering : Analysis and Design Tools

PART ATime and Frequency description of

a digital signal

2006-2007 3Dr. R.Weber / Digital Signal Processing

Temporal Descriptionx=[ 2.1 -0.5 1.3 0.5 0 1.7 -0.4 0.3 -1.9 -0.8 ];

n (time index or relative time)

0 1 2 3 4 5 6 7 8 9 10-1

What really happens here ? What really happens here ?


Theoretical spectral description

"" " " 2f f f

0 0

( ) cos(2 ) sin(2 ) j fn

f f f

x n A j fn B j fn C e

•periodicity of the spectrum •f can be limited to [0, 0.5] for real signal (spectrum symmetry)•f can be limited to [- 0.5 , 0.5] or [0 , 1] for complex signal

The problem :

( ) ( )fC FT x X f Yes, with the Fourier Transform

2( ) ( ) ( ) ,n

j fnFT x X f x n fe

The answer :

X(f+1)=X(f) Remarks :


Theoretical spectrum of basic signals

Complex exponential 2( ) . Oj f nx n A e

0.5 1 1.5-0.5-1.5 -1 0 f

x = A.*exp(2*pi*j*(0:1023)*fo);2 2( ) ( )OX f A f f

fO

sine wave ( ) .cos(2 )Ox n A j f n x = A.*cos(2*pi*(0:1023)*fo);

0.5 1 1.5-0.5-1.5 -1 0 f

2 22

( ) ( ) ( )4 4O O

A AX f f f f f

fO

( ) ( )x n n impulsex = [ 1 zeros(1,1023)];

2( ) 1X f

0.5 1 1.5-0.5-1.5 -1 0 f

( ) 0 0

(0) 1

n n

( ) 0 0

(0) 1

n n


Theoretical spectrum of basic signals2(0, )N White Gaussian Noise

x = .*randn(1, 1024);2 2( )X f

0.5 1 1.5-0.5-1.5 -1

2

0 f

General case : time and frequency are related

n

f2( )

dBX f

How will appear •a pure sine wave, •a pure impulse ?

How will appear •a pure sine wave, •a pure impulse ?

specgramdemo(x,Fs);


Pure delay :

( ) ( )y n x n 2( ) ( ). j fY f X f e

2( ) ( ) oj f ny n x n e ( ) ( )oY f X f f

Modulation or frequency shift:

( ) ( ).cos(2 )Oy n x n f n 1 1( ) ( ) ( )

2 2o oY f X f f X f f

Basic spectral properties (1)


Basic spectral properties (2) Product and Convolution:

( ) ( ) ( )y n x n h n1

0

( ) ( @ )( ) ( ) ( )Y f X H f X H f d

( ) ( @ )( ) ( ) ( )k

y n x h n x k h n k

( ) ( ) ( )Y f X f H f

Parceval relation :

1+2 2

n=- 0

Energy= ( ) ( )x n X f df


Theoretical Formulation :

2( ) ( ) ( ) ,n

j ftTF x X f x n fe

Approximations : The Discrete Fourier Transform

No digital implementation possible

1. n [0, M-1]2. Only f / f=k/M

1

0

2( ) ( ) ( ) , 0, , 1

kMM

n

j nDFT x X k x n k Me

FFT(x) = DFT(x) with M a power of 2X=fft(x);

Practical spectral description (1)


Practical spectrum (n]0,M], f k/M)

0 0.5 k/M

|X(k)|

1/M 2/M

Windowed spectrum (n]0,M])

n

n

Practical spectral description (2)

f

|X(f)|

0.50

Theoretical spectrum (n]-,+ [)M

h(n)

H(f)

Possibility to reduce distortion by using different kinds of windows, h(n)


WindowingRectangular Hamming

Hann Blackmann-Harris

WINtool sous Maltab


1 2

0

1( ) ( ) , 0, , 1

kM i nM

k

x n X k e n MM

Inverse DFT :

2M-1 12

n=0 0

( )1 1 Mean Power= ( )

M

k

X kx n

M M M

Parceval relation :

cov(x)

x=ifft(X);

DFT properties

2M-1 12

n=0 0

( ) Energy= ( )

M

k

X kx n

M

Power spectral density

2

( )( )

X kkpsd

M M


Power Spectral density(1) Spectral line case :a complex exponential or s(n) is sine wave

ˆ( ) ( ) ( )s n s n h n1

0

ˆ( ) ( @ )( ) ( ) ( )k

S k S H k S H dM

( ) ( )OS f A f f with

22

20

ˆ( ) ( )k

S k A H fM

so

DFT

When 00

kf

M

22 22 2

00

ˆ( ) (0) ( )M

n

S k A H A h n

then

What is the measured power spectral density of a sinus wave when using a classical DFT ? What is the measured power spectral density of a sinus wave when using a classical DFT ?

s=sin(2*pi*(0:1023)*20/1024); plot((abs(fft(s)).^2)/1024)


Power Spectral density (2)

12

2 0

white noise

( )M

n

h n

M

Random noise case :

Problem : each slice of M samples will give a different power spectral density estimation

Problem : each slice of M samples will give a different power spectral density estimation

50% overlap

Solution : averaging of all the local power spectral density

L samplesM samples

2ˆ( )

( ) ( )l l

S kpsd

M

Impact of the windowing

2 2

1

0

( )( ) ( )

S kpsd k H d

M M

1( ) ( )l

l

psd k psdN

1

2

0

1 ( )( ) ( )

( )lMmatlab

l

n

DFT kpsd k

N h n

psd(s,M,Fs,[h(0) h(1) … h(M-1)]) =

2( )X k

M

f=k/M


Signal Power What is the mean power ? What is the mean power ?

psd

0.5 f0- 0.5 0.5 f0

psd

1

What is the mean power of the sum of signals ?

What is the mean power of the sum of signals ?


Signal Processing goals

0

( ) ( )N

ii

y n b x n i

x(n)=s(n)+b(n) processingprocessing y(n)=s’(n)+b’(n)

Example :

s(n)

x(n)=s(n)+b(n)

Spectral point of view

Goal :•s’(n) s(n)•Power(b’(n)) 0

Temporal point of view

Signal to noise ratio : dB 10SNR =10log s

b

P

P

dB 10'

New SNR =10log s

b

P

P

PART BLinear Digital Filtering :

Analysis and Design Tools


Linear Digital filtering

x(n) FilterFilter y(n)

0 1

( ) ( ) ( )N D

i ii i

y n b x n i a y n i

recursive partnon recursive part

Basic properties :

if x1(n) FilterFilter y1(n)

if x2(n) FilterFilter y2(n)then x1(n)+x2(n) FilterFilter y1(n)+y2(n)Linearity

Temporalreproducibility

if x(n) FilterFilter y(n) then x(n-T) FilterFilter y(n-T)

Formulations : temporal view

y = filter(b,a,x);b = [ b0 b1 … bN]; a = [ 1 a1 … aD];

1.


Impulse response

x(n)

FilterFilter

y(n)

Formulation :

h = filter(b,a,x);x = [ 1 zeros(1,29)];

Finite impulse response (FIR)

( ) for k=0, , N

( ) 0 elsekh k b

h k

Infinite impulse response (IIR)

/ ( ) 0o ok h k k k

0

( ) ( )N

ii

y n b x n i

0

1

( ) ( )

( )

N

ii

D

ii

y n b x n i

a y n j

Non recursive

Recursive

Consequences :

( ) 0 0

(0) 1

n n

( ) 0 0

(0) 1

n n

(n) =(n)

The impulse response

h(n)=h(n)


Example :c=0.95 c=0.985 c=0.995

c=0.999 c=1 c=1.001

h(n)

Application to stability

Formulation :+

i=0

Stable if h(i)

Finite impulse response (FIR)( ) for k=0, , N

( ) 0 elsekh k b

h k

Always stable

Infinite impulse response (IIR) Risk of instability2( ) ( ) 2 cos(2 ) ( 1) ( 2) 0.01o oy n x n c f y n c y n f Example :

x = [ 1 zeros(1,400)];c=0.985;b= 1;a=[1 -2*c*cos(2*pi*0.01) c.^2]; h = filter(b,a,x);plot(h)

Consequences :


Convolution

0

1

0

Filter type Implementation Stability

( ) ( )

very good : N+D be careful

( )

( ) ( ) poor if N large guaranteed

N

ii

D

ii

N

ii

y n b x n i

a y n i

y n b x n i

0

( ) ( ) ( ) ( @ )( )k

y n h k x n k h x n

=

( ) ( ) ( )k

x n x k n k

Formulation :

x(n)

+

x(2)(n-2)+

x(1)(n-1)+

x(0)(n)

x(3)(n-3)

+

Filter with impulse response h(n)


-- A recursive filter can be approximated by a non recursive one

which is coherent !

Remarks :

( @ )( ) ( )h n h n

h(n)


The Z-transform( ) ( )( ) ( ) , n

n

F z ZT f z f n z z

Formulation :

useful (and simple) properties:

Given X(z)=ZT(x) and Y(z)=ZT(y)- linearity : ZT(x(n) + y(n)) = X(z) + Y(z)- convolution : Zt[(x@y)(n)] = X(z) Y(z)- ZT(x(n+1)) = z X(z) ; ZT(x(n+k)) = zk X(z)- ZT(x(n-1)) = z-1 X(z) ; ZT(x(n-k)) = z-k X(z)

These make the ZT very interesting !

0 1

( @ )( ) ( ) ( ) ( )N D

i ii i

h x n y n b x n i a y n i

( ( )) ( )ZT h n H z

Application to the filtering:ZT

ZT

0

0

0 with 1

Ni

iiD

ii

i a

b z

a z

( )

( )

Y z

X z


Stability rule

root polynomial0 1decomposion

0 1,

,( )( ) .

( )

NNi

iD Ni i

D Di

ii i

p i

o izb zH z K z

a z z

z

z

Formulation :

Im(z)

Re(z)

x

x

x

x

x

o

o

o

1

1

Instable filtero

o zerosx poles

Stable filterFilter stability if |zp,i|<1

Verify this rule with the previous example

Verify this rule with the previous example

2( ) ( ) 2 cos(2 ) ( 1) ( 2)oy n x n c f y n c y n

zplane(b,a);


Frequency response (1)

Formulation :

2( ) oj f nx n e



2 2

0

( ) at

( ) ( @ )( )

( )o o

o

j f n j f k

k

FT h f

y n x h n

e h k e

2

0

( ) ( ) ( ) j fk

k

H f FT h h k e

x = [ 1 zeros(1,1023)];c=0.985;b= 1;a=[1 -2*c*cos(2*pi*0.2) c.^2]; h = filter(b,a,x);H= fft(h);f=(0:1023)/1024;plot(f,10*log10(abs(H).^2));

2( )

dBH f

( ) ( ) ( )Y f X f H f


Simple filter design method

1. Draw the frequency response, H(f), you are looking for.2. Compute the impulse response, h(n), by applying an inverse Fourier Transform3. By truncating the length of this response, you obtain a sequence of values which

are also the non-recursive coefficients of an approximate FIR version of your filter.

H(f)

f n

h(n)FT-1

n

happrox(n)

L values

Checking :

Ripple can not be removed even with L large

Solution : other type of windowing (hamming, hanning, blackmannharris…)

rectangularwindowing


2

-2 2

z=0 0

( ) ( ) = ( ) = ZT(h) j f

kj fk j f

ek k

H f h k e h k e

Another formulation :

2

0 0

2

0 0

now ( ) so ( )

N Ni j fi

i ii iD D

i i fii i

i i

b z b eH z H f

a z a e

( )

( )( )

FT bH f

FT a

c=0.985;b= 1;a=[1 -2*c*cos(2*pi*0.2) c.^2]; %zero padding to increase frequency resolutionb=[b zeros(1,1023)]; a=[a zeros(1,1020)];H= fft(b)./fft(a);f=(0:1023)/1024;plot(f,10*log10(abs(H).^2));

[H,f

]=fr

eq

z(b

,a,1

024,1

);

2( )

dBH f



Another formulation :

2z

2

1=

1

,

z

,

=

( )( ) ( ) .

( )j f

N

D N i

i

j f

De

p

e

o i

i

zH f H z K z

z

z

z

2

1 1

1 1

2

,

,

,

,

( )( ) . .

( )

N N

i ic

j f

jD D

i i

fc

o i

p ip

o i

if

fe

eH f K K

M

z M

z

f=0

X

X

O

f=0.25

f=0.5

,1o

fM

,2o

fM,2p

fM

,1p

fMO

zplane(b,a);

2( )

dBH f



pure delay :

( ) ( )h n n

( ) ( ) ( @ )( )y n x n x h n

2( ) ( )( ) j fH f TF h f e

( ) 2Hphase f f 1

groupe_delay( ) - ( )2 Hf phase f

f

general case :

( )Hphase f

1- ( )

2 Hphase ff

Time responses for the given frequencies

grpdelay(b,a,1024,1);freqz(b,a,1024,1);

x(n) FilterFilter

2( )

dBH f

Phase response (1)


linear phase filter :0( ) 2Hphase f f

Yes, if FIR filters with symmetric or antisymmetric coefficients

23.5 samples

24 samples ( )h n

( )h n

No, if IIR

0

1

( ) ( )

( )

N

ii

D

ii

y n b x n i

a y n i

Phase response (2)


Filtered Power Spectral density

( ) .cos(2 )Ox n A j f n

x(n) FilterFilter y(n) Problem : What is the psd after filtering ?

Problem : What is the psd after filtering ?

|H(f)|

2

1

0.5 f0

2( ) (0, )x n N

f0


Attenuation

Ripple

StopbandPassband

Transition band

Filter Specifications(1)


General rules :

then

Attenuation

Ripple

Transition Band

•If or

or

0

1

( ) ( )

( )

N

ii

D

ii

y n b x n i

a y n i

Order or complexity

• For a given specification, IIR is always less complex than FIR But be careful to the stability (and the phase linearity) !

Attenuation

Ripple

Transition band

Filter Specifications (2)


Basic FIR methods : ordre 63 64 coefficients, linear phase

Bandpass :

Basic IIR methods : order 24 50 coefficients

Equiripple Least-square

Butterworthorder 12 26 coefficients

Cauer or Elliptic

2( )

dBH f

( )Hphase f

Basic Design Methods

Digital Signal Processing Introduction Course R.WEBER SP2 ESI module Traitement Numérique

Documents

Transcript of Digital Signal Processing Introduction Course R.WEBER SP2 ESI module Traitement Numérique