EEE 420 Digital Signal Processing

Instructor : Erhan A. Ince

E-mail: erhan.ince@emu.edu.tr

Web page address:

http://faraday.ee.emu.edu.tr/eeng420

Digital Signal Processing And Its Benefits By a signal we mean any variable that carries or contains some kind of information that can be conveyed, displayed or manipulated.

Examples of signals of particular interest are:

- speech, is encountered in telephony, radio, and everyday life

- biomedical signals, (heart signals, brain signals)

- Sound and music, as reproduced by the compact disc player

- Video and image,

- Radar signals, which are used to determine the range and bearing of distant targets

Attraction of DSP comes from key advantages such as :

* Guaranteed accuracy: (accuracy is only determined by the number of bits used) * Perfect Reproducibility: Identical performance from unit to unit

ie. A digital recording can be copied or reproduced several times with no loss in signal quality

* No drift in performance with temperature and age

* Uses advances in semiconductor technology to achieve:(i) smaller size (ii) lower cost (iii) low power consumption (iv) higher operating speed

* Greater flexibility: Reprogrammable , no need to modify the hardware * Superior performance

ie. linear phase response can be achieved complex adaptive filtering becomes possible

Disadvantages of DSP

* Speed and Cost

DSP designs can be expensive, especially when large bandwidth signals

are involved. ADC or DACs are either to expensive or do not have sufficient

resolution for wide bandwidth applications.

* DSP designs can be time consuming plus need the necessary resources

(software etc)

* Finite word-length problems

If only a limited number of bits is used due to economic considerations

serious degradation in system performance may result.

Application Areas

Image Processing Instrumentation/Control Speech/Audio MilitaryPattern recognition spectrum analysis speech recognition secure communications

Robotic vision noise reduction speech synthesis radar processing

Image enhancement data compression text to speech sonar processingFacsimile position and rate digital audio missile

guidanceanimation control equalization

Telecommunications Biomedical Consumer applicationsEcho cancellation patient monitoring cellular mobile phonesAdaptive equalization scanners UMTS ADPCM trans-coders EEG brain mappers digital television Spread spectrum ECG Analysis digital camerasVideo conferencing X-Ray storage/enhancement internet phone

etc.

Key DSP Operations

1. Convolution

2. Correlation

3. Digital Filtering

4. Discrete Transformation

5. Modulation

Convolution Convolution is one of the most frequently used operations in DSP. Specially in digital filtering applications where two finite and causal sequences x[n] and h[n] of lengths N1 and N2 are convolved

0

][][][][][][][kk

knxkhknxkhnxnhny

where, n = 0,1,…….,(M-1) and M = N1 + N2 -1

This is a multiply and accumulate operation and DSP device manufacturers have developed signal processors that perform this action.

Correlation There are two forms of correlation :

1. Auto-correlation

2. Cross-correlation

1. The cross-correlation function (CCF) is a measure of the similarities or shared properties between two signals. Applications are cross-spectral analysis, detection/recovery of signals buried in noise, pattern matching etc.

Given two length-N sequences x[k] and y[k] with zero means, an estimate of their

cross-correlation is given by:

,...2,1,0

00 21 n

rr

nrn

yyxx

xy

xy

Where, rxy(n) is an estimate of the cross covarience

The cross-covarience is defined as

1

0

21

0

2

1

0

1

0

][1

)0(,][1

)0(

,...2,1,0][][1

,...2,1,0][][1

N

kyy

N

kxx

nN

k

nN

kxy

kyN

rkxN

r

nkynkxN

nnkykxNnr

2. An estimate of the auto-correlation of an length-N sequence x[k] with zero mean is given by

][nxx

2,1,0,]0[

][][ nr

nrn

xx

xxxx

Digital FilteringThe equation for finite impulse response (FIR) filtering is

1

0

][][][N

k

knxkhny

Where, x[k] and y[k] are the input and output of the filter respectively and h[k] for k = 0,1,2,………,N-1 are the filter coefficients

z-1

+

z-1 z-1

+ +

z-1

y(n)

x(n)

x xxxb0 b1 b2 bN-1

Filter structureFilter structure

1

0

N

kk knxbny

A common filtering objective is to remove or reduce noise from a wanted signal.

(a) (b) (c)

(d) (e) (f)

Figure : Reconstructed bi-level text images for degradation caused by h1 and AWGN.(a) Original, (b) 2D Inverse, (c) 2D Wiener, (d)PIDD, (e) 2D VA-DF, (f) PEB-FCNRT

Discrete Transformation Discrete transforms allow the representation of discrete-time signals in the

frequency domain or the conversion between time and frequency domain representations.

Many discrete transformations exists but the discrete Fourier transform (DFT) is the most widely used one.

DFT is defined as:

NjN

n

nk eWwhereWnxkX21

0

][)(

IDFT is defined as:

10,)(1

][1

0

NnWkXN

nxN

k

kn

N

MATLAB function for DFT

function [Xk] = dft(xn)

N=length(xn);

n = 0:1:N-1; % row vector for n

k = 0:1:N-1; % row vecor for k

WN = exp(-1j*2*pi/N); % Twiddle factor (w)

nk = n'*k; % creates a N by N matrix of nk values

WNnk = WN .^ nk; % DFT matrix

Xk = (WNnk*xn' );

Matlab Function for IDFTfunction [xn] = idft(Xk)

% Computes Inverse Discrete Transform

% -----------------------------------% [xn] = idft(Xk)% xn = N-point sequence over 0 <= n <= N-1% Xk = DFT coeff. array over 0 <= k <= N-1% N = length of DFT%

N = length(Xk);n = [0:1:N-1]; % row vector for nk = [0:1:N-1]; % row vecor for kWN = exp(-j*2*pi/N); % Wn factornk = n'*k; % N by N matrix of nk valuesWNnk = WN .^ (-nk); % IDFT matrixxn = (Xk' * WNnk)/N; % row vector for IDFT values

Example

Let x[n] be a 4-point sequence

otherwise

nnx

,0

30,1][

>>x=[1, 1, 1, 1];>>N = 4; >>X = dft(x,N);>>magX = abs(X) ; >>phaX = angle(X) * 180/pi;

magX= 4.0000 0.0000 0.0000 0.0000

phaX=0 -134.981 -90.00 -44.997

Modulation

Discrete signals are rarely transmitted over long distances or stored in large quantities in their raw form.

Signals are normally modulated to match their frequency characteristic to those of the transmission and/or storage media to minimize signal distortion, to utilize the available bandwidth efficiently, or to ensure that the signal have some desirable properties.

Two application areas where the idea of modulation is extensively used are:

1. telecommunications

2. digital audio engineering

High frequency signal is the carrier

The signal we wish to transmit is the modulating signal

Three most commonly used digital modulation schemes for transmitting

Digital data over bandpass channels are:

Amplitude shift keying (ASK)

Phase shift keying (PSK)

Frequency shift keying (FSK)

When digital data is transmitted over an all digital network a scheme known

As pulse code modulation (PCM) is used.