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 (sampling – quantization)

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

0.2

0.3

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

Vo

lta

ge

[V

]

ts

-0.2

-0.1

0

0.1

0.2

0.3

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

Vo

lta

ge

[V

]

ts

Analog & digital signals

Continuous function V

of continuous variable t

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

Analog Discrete function Vk of

discrete sampling

variable tk, with k =

integer: Vk = V(tk).

Digital

-0.2

-0.1

0

0.1

0.2

0.3

0 2 4 6 8 10

time [ms]

Vo

lta

ge

[V

]

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

Sampled Signal

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Analog & digital systems

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Digital vs analog processing Digital 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 system’s 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-time DSPing

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

δ(n)=1 for n=0

Unit Step u(n)=1 for n0

δ(n)=u(n)-u(n-1)

Unit Ramp r(n)=nu(n)

δ(nTs) δ[(n-3)Τs]

nΤs past

u(nTs)

nΤs past

r(nTs)

nΤs past

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Digital system example

ms

V ANALOG

DOM

AIN

ms

V

Filter Antialiasing

k

A DIGITAL

DOM

AIN

A/D

k

A

Digital Processing

ms

V ANALOG

DOM

AIN

D/A

ms

V Filter Reconstruction

Sometimes steps missing

- Filter + A/D

(ex: economics);

- D/A + filter

(ex: digital output wanted).

General scheme

Topics of this lecture.

Digital Processing

Filter

Antialiasing

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

Antialiasing Filter

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 instantly stare at the disk if it rotates with frequency 0.5 Hz?

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The sampling theorem

A signal s(t) with maximum frequency fMAX can be recovered if sampled at frequency fS > 2 fMAX .

Condition on fS?

fS > 300 Hz

t)cos(100πt)πsin(30010t)πcos(503s(t)

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

F1 F2 F3

fMAX

Example

1

Theo*

* Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotel’nikov.

Nyquist frequency (rate) fN = 2 fMAX or fMAX or fS,MIN or 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 ambiguity in frequency domain

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Antialiasing filter

-B 0 B f

Signal of interest

Out of band noise

Out of band noise

-B 0 B fS/2

f

(a),(b) Out-of-band noise 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-sampling 1

Using spectral replications to reduce sampling frequency fS req’ments.

m

BCf2Sf

1m

BCf2

m , selected so that fS > 2B

B

0 fC

f

Bandpass signal

centered on fC

-fS 0 f

S 2f

S

f fC

Advantages

Slower ADCs / electronics

needed.

Simpler antialiasing filters.

fC = 20 MHz, B = 5MHz

Without under-sampling fS > 40 MHz.

With under-sampling fS = 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 ADC 2

)qRMS(e

inputRMS10log20idealSNR (1)

Also called SQNR

(signal-to-quantisation-noise ratio)

Ideal ADC: only quantisation error eq (p(e) constant, no stuck bits…)

eq uncorrelated with signal.

ADC performance constant in time.

Assumptions

22

FSRVT

0

dt2

ωtsin2

FSRV

T

1inputRMS

Input(t) = ½ VFSR sin( t).

12N2

FSRV

12

qq/2

q/2-

qdeqep2qe)qRMS(e

eeqq

Error value

pp((ee)) quantisation error probability density

1 q

q 2

q 2

(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

f fN

Oversampling OSR=4

Nyquist Sampler PSD

fb The oversampling process takes apart

the images of the signal band.

PSD

fN/2 0

Signal Quantization noise in

Nyquist converters

fs/2

Quantization noise in

Oversampling converters

When the sampling rate increases (4

times) the quantization noise spreads

over a larger region. The quantization

noise power in the signal band is 4 times

smaller.

frequency

PSD

FN/2 0

Signal Quantization noise

Nyquist converters

Fs/2

Quantization noise

Oversampling converters

Quantization noise

Oversampling and noise

shaping converters Spectrum at the output of a noise

shaping quantizer loop compared to

those obtained from Nyquist and

Oversampling converters.

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A discreet-time system is a device or algorithm that operates on an input sequence according to some computational procedure

Digital Systems

It may be •A general purpose computer •A microprocessor •dedicated hardware •A combination of all these

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Linear, Time Invariant Systems

N

k

k knxany0

)()(System Properties • linear •Time Invariant •Stable •Causal

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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Linearity – Superposition – 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(nω) y(n)=sin2(nω)=0.5+0.5cos(2nω)

Non-linear

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The END

Back on Tuesday

Have a nice Weekend