digital signal processing -...

→ Objectives:

1- Being able to use the following tools in order to conduct

a digital signal spectral analysis:

Digital Fourier Transform (DFT)

Power Spectral Density (PSD)

Auto and cross correlations

2- Being able to synthesize a digital filter in order to create

or modify signals.

→ Provided documents (http://thomas.perso.enseeiht.fr/teaching.html):

1- The slides

2- A detailed course and exercises with solutions

3- The texts for the practical work

4- Some elements for Matlab programming.

→ A written exam including the analog part

Nathalie Thomas

Digital Signal Processing

Master SATCOM

2018-2019

Chapter 1

Introduction

- Signal digitization

- Main digital processing tools

- Processing time / real time processing

Nathalie Thomas


Master SpaceCOM

2018-2019

Example :

x(t) = sin (2pf0 t)

time

T0 = 1/f0

Analog signal:

Defined at all times by real values

Digital signal:

Defined at discrete times and taking

amplitudes in a finite set of values.

Digitization:

Sampling + quantization

Analog signal:

Defined at all times by real values

Sampled signal:

Defined at discrete times by real

values

Example :

x(kTe) = sin (2pf0 kTe)

time

Te : sampling period

Is it possible to keep all the signal

information in the collected

samples ?

No, it’s

impossible…k = sample index

Of course it is !

On one condition :

Fe= > 2 Fmax

Note : here is a periodical sampling

Te

1

Example 1

temps

Amplitude

Signal de parole :

Obtained spectrum using the Fourier transform :

frequency- Fmax Fmax

Fourier transform of the well sampled signal:

fréquencesFe-Fe - Fmax Fmax

frequencyFe-Fe

Fe> 2Fmax

Fe-Fe

Fe< 2Fmax

Fe-Fe

Fe= 2Fmax

2Fe-2Fe

-2Fe 2Fe 3Fe-3Fe 4Fe-4Fe

When the sampling frequency is reduced, the periodizations

come closer and closer and at a moment they appear in the

signal bandwidth, generating some aliasing :

frequency

Fourier transform of the sampled signal with Fe= Fmax / 12 :

time

Energy distribution

- Fmax Fmax

- Fmax Fmax

- Fmax Fmax

image d"origine

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

image sous échantillonné d"un facteur 2

50 100 150 200 250

50

100

150

200

250

image sous échantillonné d"un facteur 4

20 40 60 80 100 120

20

40

60

80

100

120

Example 2

Original image:

512*512 pixels, quantized on 8 bits

Under sampled image with a factor of 2:


Under sampled image with a factor of 4 :


Moiré effects

Quantized signal:

Defined at all times

by a finite number of values

Example :

q

Sig

nal

Dynam

ic

time

x(t) = sin (2pf0 t)

Example (uniform quantifization by truncating on

nb = 2 bits) :

time

V1(00)

V2(01)

V3(11)

V4(10)

x(t) = sin (2pf0 t) {V1, V2, V3, V4}∈x(t) = sin (2pf0 t)

Original signal

Signal quantized

on 2 bits

The number of possible values for the signal

amplitudes after quantization is given by the number of

quantization bits.

With nb bits it is possible to code 2nb levels on the

signal dynamic. q = 2nb

Signal dynamic

The signal dynamic is divided into 2nb = 22 = 4 levels

with the same size (uniform quantifization)

and all the amplitudes belonging to the same level are

assigned the same value coded on nb=2 bits. The level

width gives what is called the quantization step q :

Is it possible to keep the same

information in the quantized signal ?

Yesss !! A well done quantization adds a noise, called

quantization noise, to the original signal

(unquantized).

It can be shown that the quantization signal to noise

ratio (SNR) is given by:

The constant depends on the considered signal, but,

watever is the signal, we can note that, when the

number of quantization bits nb increases, the signal to

noise ratio also increases quickly leading to a

transparent operation.

Tough luck but no…

Quantization is an irreversible operation.

The quantized signal will never be exactly

the same as the unquantized one.

SNRQ(dB) = 10 log = 6nb+constant

Punquantized signal

Pquantization noise

But the quantized and unquantized signals can

be very close …

time

Amplitude

→ Quantized signal on 16 bits => 216 = 65536 levels:

→ Quantized signal on nb = 4 bits => 24 = 16 levels:

Example 1

temps

Amplitude

time

image d"origine

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

Example 2Original image:

512*512,

Quantized on 8 bits

Quantification : 4 bits

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500


50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

Quantized image on 2 bitsQuantized image on 4 bits

Example 1 :

{V1, V2, V3, V4}∈x(kTe) = sin (2pf0 kTe)

temps

V1(00)

V2(01)

V3(11)

V4(10)

Te : sampling period

010101000000000000010101111111101010101010…

Digital signal:

Defined at discrete times by a finite

number of values

= sampling and quantized signal

Finaly the binary information associated to the sinusoid,

or digital sinusoid, will be given by:

image d"origine

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

Original image:

512*512,

Quantized on 8 bits


338 340 342 344 346 348 350

45

50

55

60

65

70

75

80

Example 2 :

Zoom

51

2 p

ixe

ls

512 pixels

1 gray level

among

256 (28)

Needed bandwidth(for transmission on carrier frequency)

Example 1

Fra

nce In

ter

RT

L2

Le

Mouv'

Fra

nce

Cu

ltu

re

No

sta

lgie

Skyro

ck

NR

J

Virg

in R

ad

io

RM

C

Fra

nce In

fo

Fun R

adio

BF

M

88.1

0

88.7

0

89

.10

95

.20

RT

L… … … … …

96.3

0

96

.90

99

.50

10

0.0

10

0.4

101.0

102.4

10

2.8

10

3.9

104.3

10

5.5

105.9

10

6.3

10

7.2

Example of shared bandwidth: FM band in Toulouse

87.5 MHz

frequency

108 MHzFM band

Transmission channel Bandpass

Reserved for France Info

temps

Amplitude

time

frequency

Energy distribution

Digital signal processing

Why ?

→ Construct/Modify signals:• To adapt them to a transmission channel:

modulation, filtering…

• To suppress/reduce unwanted components :

noise, some frequencies…

→ Extract some information:• Needed bandwidth to transmit the signal ?

• Defects detection …

What tools ?

→ Digital Fourier transform (DFT)

→ Auto and cross correlation functions

→ Power spectral density (PSD)

→ Digital Filters

Fourier

Transform

Example 2

time

+V

-V

0 1 1 0 0 1 0 1 1 1

0 1 1 0 0 1 0 1 1 1

+V

-V

+V

-V

time

time

fréquencesfrequency

Several

Signals

carrying

the same

binary

information

Several

Occupied

bandwidth


Why ?







What tools ?




→ Digital Filters

Different

filters

Fourier

Transform

Example 3

0 100 200 300 400 500 600 700 800 900 1000

-1

0

1

signal émis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-60

-40

-20

0

20

Normalized Frequency (p rad/sample)

Pow

er/

frequency (

dB

/rad/s

am

ple

)

spectre du signal émis

0 100 200 300 400 500 600 700 800 900 1000

-2

-1

0

1

2

signal reçu

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-30

-20

-10

0

10


Pow

er/

frequency (

dB

/rad/s

am

ple

)

spectre du signal reçu

0 100 200 300 400 500 600 700 800 900 1000

-1

-0.5

0

0.5

1

signal reçu après filtrage

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-50

0

50

Normalized Frequency (p rad/sample)P

ow

er/

frequency (

dB

/rad/s

am

ple

)

spectre du signal reçu après filtrage

Transmitted signal and

transmitted signal spectrum

Before the receiver filter

SNR = 10 dB

BER = 7.8e-04

BER = 0

Received signal and

received signal spectrum

After the receiver filter

Received signal and



Why ?







What tools ?




→ Digital Filters

Filtering

Example 4

SNR = 0 dB

BER = 0,179

BER = 7.53e-04

0 100 200 300 400 500 600 700 800 900 1000

-1

0

1

signal émis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-60

-40

-20

0

20


Pow

er/

frequency (

dB

/rad/s

am

ple

)

spectre du signal émis

0 100 200 300 400 500 600 700 800 900 1000

-4

-2

0

2

4

signal reçu

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

-10

0

10


Pow

er/

frequency (

dB

/rad/s

am

ple

)

spectre du signal reçu

0 100 200 300 400 500 600 700 800 900 1000-2

-1

0

1

2

signal reçu après filtrage

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-50

0

50


ow

er/

frequency (

dB

/rad/s

am

ple

)

spectre du signal reçu après filtrage

Transmitted signal and

transmitted signal spectrum


Received signal and



Received signal and



Why ?







What tools ?




→ Digital Filters

Filtering

Example 5Transmitted Image

Received Image


Received image


SNR = 10 dB

BER = 7.86e-04

BER = 0

Image émise

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Image reçue

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Image émise

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Image reçue

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Image émise

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Image reçue

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500


Why ?







What tools ?




→ Digital Filters

Filtering

Example 6

SNR = 0 dB

BER = 0.1581

BER = 7.5483e-04

Image émise

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Image reçue

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Image émise

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Image reçue

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Image émise

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Image reçue

50 100 150 200 250 300 350 400 450 500

100

200

300

400

500

Transmitted Image

Received Image


Received image



Why ?







What tools ?




→ Digital Filters

Filtering

Example 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-40

-20

0

20

40

Temps (s)

Sig

nal ré

el filtré

Signal réel filtré deux fois

0 5 10 15 20 25 30 35 40 45 500

2000

4000

6000

8000

Indices période de signal

T(X

)

Statistique de test, seuil considéré

T(X)

seuil

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-400

-200

0

200

400

t (secondes)

Signal réel

-1500 -1000 -500 0 500 1000 1500-50

0

50

100

Fréquences (Hz)

DSP signal réel

0.18 0.2 0.22 0.24 0.26 0.28

-200

-100

0

100

200

t (secondes)

Signal réel

-1500 -1000 -500 0 500 1000 1500-50

0

50

100

Fréquences (Hz)

DSP signal réel

Zoom

After highpass filtering to suppress the 50 Hz frequency:

Detection of higher

frequency components,

sign of potential occuring

Problems (arcs)


Why ?







What tools ?




→ Digital Filters

Example 8

Amplitude

time

Noise ?

t

Rx(t)

Not only…Autocorrelation

function

Some noise

and some signal


Why ?






• Signal detection in noise …

What tools ?




→ Digital Filters

→Fourier Transform,

→Auto and cross correlation functions,

→Power spectral density,

→ Filtering.

Digital signals

Digital tools for processing

→ What differences between the theoritical

and the digitally implemented tools ?

→ Processing time ? Real time processing ?

Basic digital processing

Processing time

Digital

processing

Examples :

→ Biased estimation for the auto autocorrelation function:

→ Digital Fourier transform (DFT):

→ Digital filtering (Finite Impulse Response filter):

Processing time

= Number of Multiplication/Accumulation (MAC)

Real Time processing : processing time < Te

Chapter 2

Digital Fourier Transform

(DFT)

Nathalie Thomas


Master SpaceCOM

2018-2019

Fourier Transform (FT) Digital Fourier Transform (DFT)

Example

%Parameters

f0=100; %cosine frequency

Fe=1000; %sampling frequency

Te=1/Fe; %sampling period

N=100; %number of generated samples

%Signal generation

x=cos(2*pi*f0*[0:Te:N*Te]);

%Signal plotting

figure; plot(x)

%Computation of the signal DFT

X=fft(x);

%visualization of the DFT modulus

figure; plot(abs(X))

With Matlab :

0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

45

50

Visualization of the signal DFT modulus:

Modulus of the signal FT:

FT of the signal:

??

11 ? 92 ?

Two cosines ?

With frequencies 11 and 92 Hz ?


1- Time sampling

2- Signal of limited duration

1+2 => Sampled and time limited signal

3- Frequency sampling

??

What impacts ??

Fourier Transform (FT) 1- FT of the sampled signal

1- Time sampling => spectrum periodization !! Shannon Condition !!

Fe > 2Fmax

Fe < 2Fmax

Aliasing

Example

1- Time sampling => spectrum periodization !! How to read the spectrum !!

Visualization on [0, Fe] :

Two diracs but only one cosine

+ -

Fe

2


Example

%Parameters





%Signal generation


%Signal plotting

figure; plot(x)


X=fft(x);

%Visualization of the DFT modulus

figure; plot(abs(X))0 20 40 60 80 100 120

0

5

10

15

20

25

30

35

40

45

50

Two cosine ? No only one !

Matlab simulation:

Visualization of the DFT modulus:

+ -


1- Time sampling => spectrum periodization !! How to read the spectrum !!

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Example

%Parameters





%Signal generation


%Signal plotting

figure; plot(x)


X=fft(x);



Matlab simulation: Signal :

1- Time sampling => spectrum periodization !! Scales (time and frequencies) !!

One period = 10 seconds ??

0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

45

50

11 92

frequencies f0=11 Hz and Fe- f0=92 Hz ??

+ -


DFT modulus:

Example

%Parameters





%Signal generation


%Signal plotting

figure; plot(x)


X=fft(x);



Matlab simulation:

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Sample index

Not time in seconds !!

Size of the vector = 101 samples

Signal:

1 period = 10 signal

samples



0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1%Parameters





%Signal generation


%Signal plotting

figure; plot([0:Te:N*Te],x)


X=fft(x);



Time (seconds)

1 period = 0,01 seconds

One period = 0,01 seconds

Example

Matlab simulation:Signal:



%Parameters





%Signal generation


%Signal plotting

figure; plot(x)


X=fft(x);



0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

45

50

Two cosine? No only one ! + -

0 101

Size of the vector = 101 samples

11th 92th

f0 =10 DFT

samples

Sample index

Not frequency in Hz !!

Example

Matlab simulation:



DFT modulus:

%Parameters





%Signal generation


%Signal plotting

figure; plot(x)


X=fft(x);


figure; plot(linspace(0,Fe,length(X)),abs(X))

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

35

40

45

50

+ -

Frequency (Hz)

0 Fe

f0 Fe-f0

f0 = 100 Hz

Two cosine? No only one !

Example

Matlab simulation:



DFT modulus:

%Parameters





%Signal generation


%Signal plotting

figure; plot(x)


X=fft(x);


figure; plot(linspace(-Fe/2,Fe/2,length(X)),fftshift(abs(X))) -500 -400 -300 -200 -100 0 100 200 300 400 5000

5

10

15

20

25

30

35

40

45

50

Frequency (Hz)

0 f0- f0 Fe/2- Fe/2

The spectrum is centered

around 0

Example

Matlab simulation:



2- Digital of limited duration [0, L]

Example

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x Truncation window

Signal of finite duration

Signal of infinite duration

2- FT of the time limited signalFourier Transform (FT)

FT of the time unlimited signal:

FT of the time limited signal * FT of the truncation window:

2- Signal of limited duration [0, L] !! Spectrum (DFT) distortion !!

Example

Fourier Transform (FT) 2- FT of the time limited signal

Example 1 Example 2Sum of two cosines with close frequencies

=> Only one cosine !!

Arising problems:

=> Only one cosine !!

Sum of two cosine with very different powers

Limited separating power Ripples around transitions

2- Signal of limited duration [0,L] !! Spectrum (DFT) distortion !!


Example

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x Truncation window of

length T : w(t)

Signal of limited

duration

Unlimited Signal FT

…

Sinus

cardinal

…

Separating power

Ripple

amplitudes

2- Signal of limited duration [0,L] !! Spectrum (DFT) distortion !!


0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x Truncation window

on N samples: w(k)

FT

……xxx… x

Dirichlet Core

Ripple

amplitudes

Example

2- TF signal échantillonné et de durée limitée

1+2- Sampled and time limited signal !! Spectrum (DFT) distortion !!

2- FT of the sampled and time limited signalFourier Transform (FT)

Signal of limited

duration

Unlimited Signal

Separating power

Sinus

cardinal

Example

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x RECTANGULAR truncation window

Signal of limited duration

Unlimited signal

rectangulaireHammingBlackman

Use other truncation

windows (weight windows) ?




-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

Tracés des TF des fenêtres de troncature

rectangulaire

Hamming

Blackman

Use other truncation windows (weight windows) ?

Examples

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Tracés des fenêtres de troncature

rectangulaire

Hamming

Blackman

Several separating power

Several amplitudes for

the associated ripples




Rectangular

Hamming

Blackman

Digital weight windows |FT| of digital weight windows

Rectangular

Hamming

Blackman

Example 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

101

fenetre naturelle

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-2

100

fenetre de Hamming

%Example1

%Parameters

f1=200; %frequency for cosine 1

f2=207; % frequency for cosine 2




%Signal generation

x1=cos(2*pi*f1*[0:Te:N*Te]);


%Rectangular window

x=x1+x2;

X_V1=fft(x,4096);

%Hamming window

w=window(@hamming,length(x));

x=(x1+x2).*w.';

X_V2=fft(x,4096);

%Plots

figure

subplot(2,1,1)

plot(linspace(0,1,4096),log10(abs(X_V1)));

title(‘Rectangular window');

subplot(2,1,2)

plot(linspace(0,1,4096),log10(abs(X_V2)));

title('Hamming window');



2- Sampled and time limited signal !! Spectrum (DFT) distortion !!


Sum of two cosines with close frequencies

Rectangular window

Hamming window

%Example 2

%Paramètres

f1=200; %frequency for cosine 1

f2=320; % frequency for cosine 2




%Signal generation


x2=0.005*cos(2*pi*f2*[0:Te:N*Te]);

x=x1+x2;

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100

fenetre naturelle

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

100

fenetre de Hamming

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100

fenetre de Blackman

Example 2Sum of two cosine with very different powers

%Rectangular window

X=fft(x,4096);

figure

subplot(3,1,1)

semilogy(linspace(0,1,4096),abs(X));

axis([0 1 10^-5 10^5])

title(‘Rectangular window');

%Hamming window

w=window(@hamming,length(x));

x=(x1+x2).*w.';

X=fft(x,4096);

subplot(3,1,2)


title('Hamming window');

%Blackman window

w=window(@blackman,length(x));

%w=blackman(N);

x=(x1+x2).*w.';

X=fft(x,4096);

subplot(3,1,3)


title('Blackman window');





Rectangular window

Hamming window

Blackman window

Example 3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100

Fréquences normalisées

|X(f

)|

fenetre naturelle

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-2

100

102


|X(f

)|

fenetre de Hamming

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100


|X(f

)|

fenetre de Blackman

Signal ???




Rectangular window

Hamming window

Blackman window

Normalized frequency



Example 3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100


|X(f

)|

fenetre naturelle

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-2

100

102


|X(f

)|

fenetre de Hamming

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100


|X(f

)|

fenetre de Blackman

3 cosines

with the same power

Proposed solution: Use SEVERAL truncation (or weight) windows for one analysis

2 cosines

with the same power

but something is strange…

3 cosines

with different powers

+ -

4 cosines:

3 with the same power

1 with a lower power




Rectangular window

Hamming window

Blackman window




Fourier Transform (FT)

3- Frequency sampling

Example

%Parameters





%Signal generation


%Signal plotting

figure; plot(x)


X=fft(x);


figure;

plot(linspace(0,Fe,length(X)),abs(X))

Matlab simulation:

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

35

40

45

50

Two cosine ? No only one !

Plot of the DFT modulus:

+ -

Frequency (Hz)

0 Fe

f0 Fe-f0

But where are

the Dirichlet cores ???


Example

DFT modulus

f (Hz)f0 Fe-f00

Fe/N

f (Hz)f0 Fe-f00

Fe/N

DFT modulus


3- Frequency sampling !! Bad spectrum visualization !!


Mdulus of X(n)

f (Hz)f0 Fe-f00

Fe/N

Proposed solution: interpolation with Zero Padding

xx

x

x

x

x

x

xx

x

x

x

xx

xx

xxx

xxxx

xxxx

x

xxxx

xx

xx

x

x

x

x

x

N points spaced from Fe/N between 0 and Fe

Example

Modulus of Y(n)

f (Hz)f0 Fe-f00

Fe/N

xx

x

x

x

x

x

xx

x

x

x

xx

xx

xxx

xxxx

xxxx

x

xxxx

xx

xx

x

x

x

x

x

Fe/MN

MN points spaced from Fe/MN between 0 and Fe




0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

35

40

45

50

Exemple

Two cosine? No only one !

DFT modulus:

+ -

Frequency (Hz)

0 Fe

f0 Fe-f0

%Parameters





%Signal generation


%Signal plotting

figure; plot(x)


X=fft(x,128);


figure;

plot(linspace(0,Fe,length(X)),abs(X))

Here are

the Dirichlet cores !





Matlab simulation:

0 500 10000

20

40

60

Fréquences (Hz)

|TF

D(x

)|

sans ZP

0 500 10000

20

40

60

Fréquences (Hz)

|TF

D(x

)|

ZP : 128

0 500 10000

20

40

60

Fréquences (Hz)|T

FD

(x)|

ZP : 256

0 500 10000

20

40

60

Fréquences (Hz)

|TF

D(x

)|

ZP : 512

Example

% Plots of the DFT modulus

figure;

subplot(2,2,1)

plot(linspace(0,Fe,length(X1)),abs(X1))

xlabel(‘Frequency (Hz)’)

subplot(2,2,2)



subplot(2,2,3)



subplot(2,2,4)



Plots of the DFT modulus

For several value of the Zero Padding parameter:

%Parameters





%Signal generation


%Signal plotting

figure; plot([0:Te:N*Te],x)

%Computation of the DFT

X1=fft(x);

X2=fft(x,128);

X3=fft(x,256);

X4=fft(x,512);

Zero Padding

parameter





Matlab simulation:no

Frequency (Hz)

Frequency (Hz) Frequency (Hz)

Frequency (Hz)

3- Frequency sampling => signal periodization

!! A product is changed into a circular convolution product by DFT or DFT-1 !!

Inverse Fourier transform (FT-1) Digital inverse Fourier transform(DFT-1)

DFT DFT-1

x(k) is periodical of period N

Linear (« classical ») convolution:

Circular convolution:



Linear convolution: Circular convolution:

. . . 0 0 0 1 2 3 0 0 0 . . .

. . . 0 0 0 1 1 1 0 0 0 . . .. . . 0 0 0 1 1 1 0 0 0 . . .

. . . 0 0 0 1 1 1 0 0 0 . . .. . . 0 0 0 1 1 1 0 0 0 . . .

. . . 0 0 0 1 1 1 0 0 0 . . .. . . 0 0 0 1 1 1 0 0 0 . . .

. . . 0 0 0 1 1 1 0 0 0 . . .

0136530

x1(p) :

x2(k-p) : p=-3p=-2p=-1p=0p=1p=2p=3

. . . 0 0 0 1 1 1 0 0 0 . . .x2(p) :. . . 1 2 3 1 2 3 1 2 3 . . .

. . . 1 1 1 1 1 1 1 1 1 . . .. . . 1 1 1 1 1 1 1 1 1 . . .

. . . 1 1 1 1 1 1 1 1 1 . . .. . . 1 1 1 1 1 1 1 1 1 . . .

. . . 1 1 1 1 1 1 1 1 1 . . .. . . 1 1 1 1 1 1 1 1 1 . . .

. . . 1 1 1 1 1 1 1 1 1 . . .

6666666

p=-3p=-2p=-1p=0p=1p=2p=3

. . . 1 1 1 1 1 1 1 1 1 . . .

x1(pmodulo N) :

x2(pmodulo N) :

Example (N=3)


3- Frequency sampling => signal periodization !! Circular convolution = linear convolution !!

Linear convolution = circular convolution

. . . 0 0 0 1 2 3 0 0 0 . . .

. . . 0 0 0 1 1 1 0 0 0 . . .. . . 0 0 0 1 1 1 0 0 0 . . .

. . . 0 0 0 1 1 1 0 0 0 . . .. . . 0 0 0 1 1 1 0 0 0 . . .

. . . 0 0 0 1 1 1 0 0 0 . . .. . . 0 0 0 1 1 1 0 0 0 . . .

. . . 0 0 0 1 1 1 0 0 0 . . .

0136530

. . . 0 0 0 1 1 1 0 0 0 . . .

01

6530

. . . 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 . . .

. . . 1 2 3 0 0 0 1 2 3 0 0 0 1 2 3 0 0 0. . .

. . . 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 . . .. . . 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 . . .

. . . 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 . . .. . . 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 . . .

. . . 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 . . .. . . 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 . . .

. . . 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 . . .. . . 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 . . .

3

0

By adding N zeros

Example (N=3)


3- Frequency sampling => signal periodization


Digital Fourier Transform (DFT)Inverse Digital Fourier Transform (DFT-1)

4- DFT properties:

→ Linearity

→ Translation => phase rotation

→ Hermitic symetry

→ Circular convolution

→ Parseval equality

→ Fast Fourier Transform Algorithm (FFT) for a lower computation time: Nlog2(N)<<N2 for a DFT (or DFT-1) on N points

5- An algorithm to reduce the computation time : Cooley Tuckey FFT (Fast Fourier Transform)

Even indexes

(DFT of order N/2)

…

DFT of order N = 2p : N2 (+/x) operations

First decomposition: N+2(N/2)2 << N2 (+/x) operations

Second decomposition: 2(N/2)+4(N/4)2 (+/x) operations

Odd indexes

DFT of order N/2

4 DFTs of order N/4

Np DFT of order 2 = N log2(N) (+/x) operations

Even indexes Odd indexes Even indexes Odd indexes


N (+/x) operations

http://1-open-opensocial.googleusercontent.com/gadgets/ifr?v=4c2f39d3edfbfbfd48696c2f72221a5e&container=open&view=home&mid=1&lang=all&url=http://www.sitmo.com/gg/latex/latex.xml&country=ALL&source=http://www.sitmo.com/latex/&parent=http://www.sitmo.com/latex/&libs=core:core.io:rpc




→ Time sampling => spectrum periodization- Respect Shannon condition

- Pay attention to the way you read the spectrum and the scales (in time and frequency)

→ Signal of limited duration => spectrum distortion- Use several weight windows for one signal analysis

=> several separating powers and amplitudes for the ripples appearing in the frequency domain

→Frequency sampling => signal periodization- Pay attention to the lecture of the spectrum

=> use the zero padding interpolation technique to have a good visualization

- DFT and DFT-1 convert a product into a circular convolution product

=> if a linear convolution is required: linear convolution = circular convolution thanks to zero padding

→FFT = Fast Fourier Transform algorithm- Condition N=2p => decomposition of the DFT into interlaced subseries

- Processing time: Nlog2(N) << N2


Chapter 3

Digital auto and cross

correlation functions

Nathalie Thomas


Master SpaceCOM

2018-2019

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1-60

-40

-20

0

20

40

60

Temps (s)

Rx

Example

%Parameters

f0=100; %frequency of the cosine




%Signal generation

x=cos(2*pi*f0*[0:Te:N*Te]+rand*2*pi);

% Computation and plotting of the biased

% autocorrelation

Rx=xcorr(x);

figure; plot([-N*Te:Te:N*Te],Rx);

xlabel('Time (s)'); ylabel('R_x')

Matlab simulation: ??

(Hermitic symetry)

Time (s)

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1-60

-40

-20

0

20

40

60

Temps (s)

Rx

Multiplicative

Triangular bias

BIAISED estimator

Example

%Parameters





%Signal generation

x=cos(2*pi*f0*[0:Te:N*Te]+rand*2*pi);

% Computation and plotting of the biased

% autocorrelation

Rx=xcorr(x);

figure; plot([-N*Te:Te:N*Te],Rx);


Matlab simulation:

(Hermitic symetry)

Time (s)

UNBIASED estimator

High variance

on the sidesMatlab simulation:

%Parameters





% Computation and plotting of the unbiased

% autocorrelation for several signal realizations

x1=cos(2*pi*f0*[0:Te:N*Te]+rand*2*pi);

Rx1=xcorr(x1,'unbiased');





figure

plot(Rx1); hold on; plot(Rx2,'r'); plot(Rx3,'g')

legend('realization 1','realization 2','realization3')


Example

0 20 40 60 80 100 120 140 160 180 200

-0.5

0

0.5

1

Temps (s)R

x

réalisation 1

réalisation 2

réalisation3

zooms

175 180 185 190 195 200 205 210-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Temps (s)

Rx

réalisation 1

réalisation 2

réalisation3

2 4 6 8 10 12 14

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Temps (s)R

x

réalisation 1

réalisation 2

réalisation3

(Hermitic symetry)

Time (s)

Time (s)Time (s)

realization 1

realization 2

realization 3

1 – Time domain estimation (biased or unbiased)

2- Frequency domain estimation

Processing time ~ N2 (+/x) operations

Linear convolution:

Processing time for the computation of a digital cross correlation function

If:

And:

then:

Processing time: N+3Nlog2(N) (+/x) operations

The frequency domain estimation allows to reduce the pocessing time:

N+3Nlog2(N) << N2 (+/x) operations

DFT

BUT

!! Signals are periodical !!

If:

- we can put an infinite sum in the expression of Rxy(k) instead of 0,…, N-1

- the DFT converts a linear convolution product into a product

Chapter 4

Power Spectral Density

(PSD)

Nathalie Thomas


Master SpaceCOM

2018-2019

→ Periodogram

→ Correlogram

→ Drawback: inconsistent estimators

• Convolutive bias:

• The variance does not depend on the signal observation length:

Two basic estimators

Note: périodogram biased correlogram

Fejer core :

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

20

40

60

80

100

120

140

Frequences normaliséesNormalized frequency

Examples:

0 100 200 300 400 500 600 700 800 900 1000-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Temps

x(t)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


DS

P

DSP simulée

DSP théorique

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Temps

x(t)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


DS

P

DSP simulée

DSP théorique10 000 signal samples

1 000 signal samples

Two basic estimators



PS

DP

SD

Simulated PSD

Theoretical PSD

Simulated PSD

Theoretical PSD

Modified versions

Objective: reduce the estimation variance when the signal observation length increases

→ Cumulative periodogram (Bartlett) :

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Temps

x(t)

…

LN=ML

→ Cumulative periodogram (Bartlett) :

Examples :

Drawback: for a given signal observation length, the bias increases when the variance is reduced

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7


DS

P

DSP simulée

DSP théorique

N= 10 000 samples, 1 window

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


DS

P

DSP simulée

DSP théorique

N= 10 000 samples, M=10 windows

of L= 1000 samples

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4


DS

P

DSP simulée

DSP théorique

N= 100 000 samples, M=100 windows

of L=1 000 samples

Example :

(N=100, M=1) => (N=100, M=10, L=10)

Fejer core:

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequences normalisées

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

80

90

100

Frequences normalisées

100 samples 10 samples

2/N

Modified versions


PS

D


PS

D


PS

D

Normalized frequency Normalized frequency

Simulated PSD

Theoretical PSDSimulated PSD

Theoretical PSD

Simulated PSD

Theoretical PSD

→ Modified periodogram:

→ Modified correlogram:

Drawback: abrupt variations are smoothed, the separating power is reduced

→ Welch periodogram = cumumative and modified periodogram

- Sliding window => M’ > M windows of length L allowed to overlap

- Modified periodogram on each signal section

- Example:

Weigth window

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4


DS

P

DSP simulée

DSP théorique

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4


DS

P

DSP simulée

DSP théorique

N= 100 000 samples

M=100 windows

of L=1 000 samples

M=100 windows of L=1 000 samples

Overlapping = L/2, Hamming windowing

Modified versions

Simulated PSD

Theoretical PSD

Simulated PSD

Theoretical PSD


PS

D

PS

D

→ Examples of cumulative periodograms:

→ Welch periodogram:(overlapping = L/2)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7


DS

P

DSP simulée

DSP théorique

K= 10 000 samples,

1 window

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


DS

P

DSP simulée

DSP théorique

K= 10 000 samples, M=10 windows

of L= 1000 samples

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4


DS

P

DSP simulée

DSP théorique

K= 100 000 samples, M=100 windows

of L=1 000 samples

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


DS

P

DSP simulée

DSP théorique

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4


DS

P

DSP simulée

DSP théorique

Modified versions

Simulated PSD

Theoretical PSD

Simulated PSD

Theoretical PSDSimulated PSD

Theoretical PSD

Simulated PSD

TheoreticalPSDSimulated PSD

Theoretical PSD




PS

D PS

D

PS

D

PS

D

PS

D

Chapter 5

Digital Filtering

Nathalie Thomas


Master SpaceCOM

2018-2019

→ Definition :

→ Analysis of ANALOG linear time invariant systems

o Time analysis (step response, ramp response): stability (poles of H(p) with negative real parts), rapidity (rise time, settling

time), precision (static error, following error)

o Frequency analysis (response to a sinusoidal input) : Bode diagrams=> cutoff frequency, bandpass, stopband attenuation,

resonnance …

o Example :

-90°w0

Step response

Analog linear time invariant filters

A TOOL FOR THEIR STUDY: THE LAPLACE TRANSFORM

Reminders

Transfer function

Frequency

response

Bode diagram

→ Definition:

→ Analysis of DIGITAL linear time invariant systems

o Main properties:

• Linearity:

• Time shift:

• Convolution product:

o Existence:

o inversion:

o Note: tables of ZT exist

Digital linear time invariant filters

A TOOL FOR THEIR STUDY: THE Z-TRANSFORM

→ Linearity:

→ Time invariance:

→ Impulse response and transfer function:

→ Frequency response and group propagation time (GPT)

Note:

Filter

Filter

Filter

Filter Filter

Filter

Impulse

response

Transfer fonction

Digital impulse

(Kronecker symbol)

Frequency

responseGPT


DEFINITIONS

→ Three conditions on the impulse response:

o Causality:

o Stability:

o Real impulse response:


REALISABILITY

→ By analogy with analog filters:

Example :

→ Digital filters definition:

→ IIR filters = INFINITE IMPULSE RESPONSE filters:

x(t) y(t)

R

C

i(t)

+

Differential equation in the time domain

Rationnal transfer function in p-domain

RATIONNAL transfer function in z-domain

Recurrence equation in the time domain

(Note: )

Closed-loop system

%filtering with Matlab

y=filter(B,A,x);

Definition of the

wanted filter (two sets

of coefficients)

Filtered signal Signal to filter


RATIONNAL FILTERS

Filtres numériques linéaires invariants dans le temps

STABILITE DES FILTRES RATIONNELS DE TYPE RII

→ Condition on the poles of the transfer function:

→ Condition on the coefficients :

o 1st order filters

o 2nd order filters

(hypothesis : N<M)

(causal solution(1))

(1)) The inverse ZT is not unique. It will be different depending on the chosen closed curve to compute it: see exercises (http://thomas.perso.enseeiht.fr/teaching.html).

Stability condition

for the digital rationnal filters

1

- 1

- 1 1 2- 2

Stable filter

Unstable filter

Stability triangle

for 2nd order IIR filters

Stability condition for the 1st order IIRs

Stability condition for the 2nd order IIRs


IIR STABILITY

→ Defined by:

o Always stables

o Only defined by one set of coefficients bk, k=0, …, N-1

Non-recursive filters

%Filtering with Matlab

y=filter(B,1,x);

Definition of the wanted

FIR filter (only one

set of coefficients)

Filtered signal Signal to filter


FINITE IMPULSE RESPONSE FILTERS (FIR)

Specifications to follow:

TEMPLATE

COEFFICIENTS

defining a filter compliant with the given specifications

SYNTHESIS

IMPLANTATION on Matlab : IMPLANTATION in real time:

y=filter(B,A,x);

Filtered signalSignal to filter

Sets of coefficients

defining the filter to be used

IIR filter

To be computed in Te seconds

(sampling period)

Note: A=[1] for FIR filters

FIR filter


RATIONAL FILTERS: SYNTHESIS AND IMPLEMENTATION

Low-pass filter High-pass filter Band-pass filter Notch filter

→ Ideal target frequency responses:

→ Template to follow:

o On the modulus of the frequency response

Examples :

o On the phase of the frequency response: phase or all-pass filter.

BandpassCutted

bandwidth

Transition

area

BandpassBandpassCutted

bandwidth

Transition

area

Transition

area

Cutted

bandwidth

LOW-PASS filter HIGH-PASS filter


THE TEMPLATE

Transition

area

Low-pass filter High-pass filter Band-pass filter Notch filter

→ Ideal target frequency responses:

→ Template to follow:

o On the modulus of the frequency response

Examples :

o On the phase of the frequency response: phase or all-pass filter.


THE TEMPLATE

Bandpass

Cutted

bandwidth

Transition

area

Transition

area

Cutted

bandwidth

Low-pass filter

LINEAR SCALE

Bandpass Cutted

bandwidth

Transition

area

Low-pass filter

LOGARITHM SCALE (dBs)

COEFFICIENTS

OF A FILTER following the given specifications

IIR filter

To be computed in Te seconds

(sampling period)

FIR filter

There is not only one filter

following the template

FIR synthesis

IIR synthesis

Two different

methods


RATIONAL FILTERS: SYNTHESIS AND IMPLEMENTATION


TEMPLATE

SYNTHESIS

IMPLANTATION on Matlab : IMPLANTATION in real time:

y=filter(B,A,x);

Filtered signalSignal to filter

Sets of coefficients

defining the filter to be used

Note: A=[1] for FIR filters

Specifications:

TEMPLATE

COEFFICIENTS

SYNTHESIS

→ Example: DIGITAL low-pass filter

+

+

+

+

++

++

++

++

+

+

+

++

++

++

++

Frequency periodisation

Sampling

Signal limitation

to N coefficients

Distortions :- Transitions are smoothed

- Ripples appear around

FILTER ORDER


FIR SYNTHESIS

+

+

+

+

++

++

++

++

+

+

+

++

++

++

++

+

+

+

+

++

++

++

++

+

+

+

++

++

++

++

Introduction of a delay

Non causal filter:

Causal filter:(Hyp : N is odd)

→ Example: DIGITAL low-pass filter


FIR SYNTHESIS

Specifications:

TEMPLATE

COEFFICIENTS

SYNTHESIS

0 50 100 150 200 250

-1

0

1

signal filtré

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-60

-40

-20

0

20


Pow

er/

frequency (

dB

/rad/s

am

ple

)

spectre du signal filtré

+

+

+

+

++

++

++

++

+

+

+

++

++

++

++

0 20 40 60 80 100 120 140 160 180

-1

0

1

signal filtré

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-60

-40

-20

0

20


ow

er/

frequency (

dB

/rad/s

am

ple

)

spectre du signal filtré

+

+

+

+

++

++

++

++

+

+

+

++

++

++

++

Introduction of a delay between the input

and the output signals:

Introduction of a delay

in the impulse response:

+++

++++

+++++

++++

++++++++

(Hyp : odd order)

(Hyp: N is odd)

(Hyp: N is odd)

(order = 81)

(order = 201)

→ Example: CAUSAL DIGITAL low-pass filter


FIR SYNTHESIS

Filtered signal

Filtered signal

+

+

+

+

++

++

++

++

+

+

+

++

++

++

++

(Hyp : N is odd)

Constant if h(n) odd or even

The GPT for FIR filters

is constant

If h(n) is odd or even

A constant value is added to the GPT:

→ Example: CAUSAL DIGITAL low-pass filter


FIR SYNTHESIS

→ Parameters allowing to follow the template: ORDER and WEIGTH WINDOW

-200 -150 -100 -50 0 50 100 150 200 250

0.2

0.4

0.6

0.8

1

1.2

Fréquences (Hz)

H(f

)

Ordre = 51

Ordre = 101

Ordre = 201

-500 -400 -300 -200 -100 0 100 200 300 400 50010

-8

10-6

10-4

10-2

100

102

Fréquences (Hz)

H(f

)

Fenêtre rectangulaire

Fenêtre de Hamming

Fenêtre de Blackman

→ Order influence

(rectangular window)→ Truncation window influence

(order = 21)

Transitions are more

or less attenuatedThe ripples amplitudes

are more or less important


FIR SYNTHESIS

Frequency (Hz) Frequency (Hz)

Order = 51

Order = 101

Order = 201

Rectangular window

Hamming window

Blackman window

Analog patterns library:Butterworth, Tchebycheff,

Cauer, Bessel…


H ( f )

Transfer function:

H ( p )


H ( f )~

Transfer function:

H ( z )

?

?

𝑓 = ሚ𝑓𝐹𝑒

BILINEAR TRANSFORM:

The stability and the frequency response are kept

𝐻 𝑧 = 𝐻(𝑝)𝑝=

2𝑇𝑒

1−𝑧−1

1+𝑧−1

ሚ𝑓 =1

𝜋arctan(𝜋𝑓𝑇𝑒)

BUT the frequency axis is distorded:

𝑓 =1

𝜋𝑇𝑒tan(𝜋 ሚ𝑓)

Predistorsion

→ An example of IIR synthesis is given in the exercises document (http://thomas.perso.enseeiht.fr/teaching.html)


IIR SYNTHESIS

Specifications:

TEMPLATE

COEFFICIENTS

SYNTHESIS

→ Direct structure → Canonic structure

M+N+1 (+/x) operations, 2 delayed lines

M+N+1 (+/x) operations, 1 delayed line


RATIONAL FILTERS IMPLANTATION

→ Decomposed structures

→ Series (or cascade) :

→ Non recursive structure (FIR)

Hi(z) : first or second order cells

→ Parallel :


RATIONAL FILTERS IMPLANTATION

To go further

Some references:

→ Digital Signal Processing: Course and exercices with solutions, by Nathalie Thomas,

ENSEEIHT documentation : http://thomas.perso.enseeiht.fr/teaching.html

→ Signal and Systems, by Simon Haykin and Barry Von Veen, Wiley

→ Digital Signal Processing, by Alan V. Oppenheim, Ronald W. Schafer, Prentice-Hall.

→ Documents on complex variable, Laplace transform, z tranform :

http ://dobigeon.perso.enseeiht.fr/teaching.html

http://thomas.perso.enseeiht.fr/teaching.html