1 Basics DSP AV Intro

8/10/2019 1 Basics DSP AV Intro

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Basics on Digital Signal Processing

Introduction

Vassilis Anastassopoulos

Electronics Laboratory, Physics Department,

University of Patras


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Outline of the Course

1. Introduction (samplingquantization)

2. Signals and Systems

3. Z-Transform

4. The Discreet and the Fast Fourier Transform

5. Linear Filter Design

6. Noise

7. Median Filters


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-0.2

-0.1

0

0.1

0.20.3

0 2 4 6 8 10sampling time, tk [ms]

Voltage[V]

ts

-0.2

-0.1

0

0.1

0.20.3

0 2 4 6 8 10sampling time, tk [ms]

Voltage[V]

ts

Analog & digital signals

Continuous functionV

of continuousvariable t

(time, space etc) : V(t).

Analog

Discrete functionVkofdiscretesampling

variable tk, with k =

integer: Vk = V(tk).

Digital

-0.2

-0.1

0

0.1

0.20.3

0 2 4 6 8 10

time [ms]

Voltage[V]

Uniform (periodic) sampling.Sampling frequency fS= 1/ tS

SampledSignal


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Analog & digital systems


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Digital vsanalog processingDigital Signal Processing (DSPing)

More flexible.

Often easier system upgrade.

Data easily stored -memory.

Better control over accuracy

requirements.

Reproducibility.

Linear phase

No drift with time and

temperature

Advantages

A/D & signal processors speed:

wide-band signals still difficult to

treat (real-time systems).

Finite word-length effect.

Limitations


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DSPing: aim & tools

Software Programming languages: Pascal, C / C++ ...

High level languages: Matlab, Mathcad, Mathematica

Dedicated tools (ex: filter design s/w packages).

Applications Predicting a systems output.

Implementing a certain processing task.

Studying a certain signal.

General purpose processors (GPP), -controllers.

Digital Signal Processors (DSP).

Programmable logic ( PLD, FPGA ).

Hardware real-timeDSPing

Fast

Faster


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Related areas


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Applications


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Important digital signals

Unit Impulse or Unit Sample.

The most important signal for

two reasons

(n)=1 for n=0

Unit Step u(n)=1 for n0(n)=u(n)-u(n-1)

Unit Ramp r(n)=nu(n)

(nTs) [(n-3)s]

ns past

u(nTs)

ns past

r(nTs)

ns past


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Digital system example

ms

V ANALOG

DOMAIN

ms

VFilter

Antialiasing

k

A D

IGITAL

D

OMAIN

A/D

k

A

DigitalProcessing

ms

V ANALOG

DOMAIN

D/A

ms

V FilterReconstruction

Sometimes steps missing

- Filter + A/D

(ex: economics);- D/A + filter

(ex: digital output wanted).

General scheme

Topics of thislecture.

DigitalProcessing

FilterAntialiasing

A/D


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Digital system implementation

Sampling rate.

Pass / stop bands.

KEY DECISION POINTS:

Analysis bandwidth, Dynamic range

No. of bits. Parameters.

1

2

3

Digital

Processing

A/D

AntialiasingFilter

ANALOG INPUT

DIGITAL OUTPUT

Digital format.What to use for processing?


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AD/DA Conversion General Scheme


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AD Conversion - Details


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Sampling


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Sampling

How fast must we sample a continuous

signal to preserve its info content?

Ex: train wheels in a movie.

25 frames (=samples) per second.

Frequency misidentification due to low sampling frequency.

Train starts wheels go clockwise.

Train accelerates wheels go counter-clockwise.

1

Why?


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Rotating Disk

How fast do we have to instantlystare at the disk if it rotateswith frequency 0.5 Hz?


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The sampling theorem

A signal s(t) with maximum frequency fMAX

can berecovered if sampled at frequency fS> 2 fMAX.

Condition on fS?

fS> 300 Hz

t)cos(100t)sin(30010t)cos(503s(t)

F1=25 Hz, F2= 150 Hz, F3= 50 Hz

F1 F2 F3

fMAX

Example

1

Theo*

*Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotelnikov.

Nyquist frequency (rate) fN= 2 fMAXor fMAXor fS,MINor fS,MIN/2

Naming gets

confusing !


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Sampling and Spectrum


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Sampling low-pass signals

-B 0 B f

Continuous spectrum(a) Band-limited signal:

frequencies in [-B, B] (fMAX= B).

(a)

-B 0 B fS/2 f

Discrete spectrum

No aliasing (b) Time sampling frequency

repetition.fS> 2 B no aliasing.

(b)

1

0 fS/2 f

Discrete spectrum

Aliasing & corruption(c)

(c) fS 2 B aliasing !

Aliasing: signal ambiguityin frequency domain


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Antialiasing filter

-B 0 B f

Signal of interest

Out of bandnoise Out of bandnoise

-B 0 B fS/2

(a),(b)Out-of-bandnoise can aliase

into band of interest. Filter it before!

(a)

(b)

(c)

Passband: depends on bandwidth of

interest.

Attenuation AMIN: depends on

ADC resolution ( number of bits N).

AMIN, dB~ 6.02 N + 1.76

Out-of-band noise magnitude.

Other parameters: ripple, stopband

frequency...

(c)Antialiasing filter

1


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Under-sampling1

Using spectral replications to reduce

sampling frequency fSreqments.

m

BCf2Sf

1m

BCf2

m , selected so that fS> 2B

B

0 fC

Bandpass signal

centered on fC

-fS 0 f

S 2f

S

fC

Advantages

Slower ADCs / electronics

needed.

Simpler antialiasing filters.

fC = 20 MHz, B = 5MHz

Without under-samplingfS> 40 MHz.

With under-samplingfS= 22.5 MHz (m=1);

= 17.5 MHz (m=2); = 11.66 MHz (m=3).

Example


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Quantization and Coding

q

N Quantization Levels

Quantization Noise


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SNR of ideal ADC2

)qRMS(e

inputRMS

10log20idealSNR(1)

Also called SQNR

(signal-to-quantisation-noise ratio)

Ideal ADC: only quantisation erroreq(p(e)constant, no stuck bits)

equncorrelated with signal.

ADC performance constant in time.

Assumptions

22

FSRVT

0

dt2

tsin2

FSRV

T

1inputRMS

Input(t) = VFSRsin(t).

12N2

FSRV

12

qq/2

q/2-

qdeqep2qe)qRMS(e

eeqqError value

pp((ee))quantisation error probability density

1q

q2

q2

(sampling frequency fS= 2 fMAX)


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SNR of ideal ADC - 2

[dB]1.76N6.02SNRideal (2)Substituting in (1) :

One additional bit SNR increased by 6 dB

2

Actually(2) needs correction factor depending on ratio between sampling freq

& Nyquist freq. Processing gain due to oversampling.

- Real signals have noise.

- Forcing input to full scale unwise.

- Real ADCs have additional noise (aperture jitter, non-linearities etc).

Real SNR lower because:


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Coding - Conventional


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Coding Flash AD


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DAC process


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Oversampling Noise shaping

fs=4fN(b)

(a)

f

ffN

Oversampling OSR=4

Nyquist SamplerPSD

fb The oversampling process takes apart

the images of the signal band.

PSD

fN/20

SignalQuantization noise in

Nyquist converters

fs/2

Quantization noise in

Oversampling converters

When the sampling rate increases (4times) the quantization noise spreads

over a larger region. The quantization

noise power in the signal band is 4 times

smaller.

frequency

PSD

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SignalQuantization noise

Nyquist converters

Fs/2

Quantization noise

Oversampling converters

Quantization noise

Oversampling and noiseshaping converters

Spectrum at the output of a noiseshaping quantizer loop compared to

those obtained from Nyquist and

Oversampling converters.


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A discreet-time system is a device or algorithmthat operates on an input sequence according tosome computational procedure

Digital Systems

It may be

A general purpose computerA microprocessordedicated hardwareA combination of all these


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Linear, Time Invariant Systems

N

k

k knxany0

)()(

System Properties

linearTime InvariantStableCausal

Convolution


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Linear Systems - Convolution

5+7-1=11 terms


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Linear Systems - Convolution

5+7-1=11 terms


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L

k

k

M

k

k knybknxany10

)()()(

General Linear Structure


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Simple Examples


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LinearitySuperposition Frequency Preservation

Principle of Superposition

H

y1(n)x1(n)

H

y2n)x2(n)

H

ay1(n)+by2(n)ax1(n)+bx2(n)

Principle of Superposition Frequency Preservation

x2

x12(n)x1(n)

x2(n)

x1(n)+x2(n)

x2

x2

x22(n)

x12(n)+x2

2(n)+2 x1(n) x2(n)

If y(n)=x2(n) then for x(n)=sin(n)y(n)=sin2(n)=0.5+0.5cos(2n)

Non-linear


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The END

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