EEE 420 Digital Signal Processing

Instructor : Erhan A. Ince E-mail: [email protected]

Web page address: http://faraday.ee.emu.edu.tr/eee420 http://faraday.ee.emu.edu.tr/eaince

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 Military Pattern recognition spectrum analysis speech recognition secure communications Robotic vision noise reduction speech synthesis radar processing Image enhancement data compression text to speech sonar processing Facsimile animation position and rate control digital audio missile guidance equalization Telecommunications Echo cancellation Adaptive equalization ADPCM trans-coders Spread spectrum Video conferencing Biomedical patient monitoring scanners EEG brain mappers ECG Analysis X-Ray storage/enhancement Consumer applications cellular mobile phones UMTS digital television digital cameras internet phone etc.

Key DSP Operations 1.

2.

3.

4.

5.

Convolution Correlation Digital Filtering Discrete Transformation 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

N

1 convolved and

N

2 are

y

[

n

] 

h

[

n

] 

x

[

n

] 

k

   

h

[

k

]

x

[

n



k

] 

k

   0

h

[

k

]

x

[

n



k

] where, n = 0,1,…….,(M-1) and M =

N

1 +

N

2 -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: 

xy

 

r xx

0

r xy r yy

 

1  2

n

 0 ,  1 ,  2 ,...

Where,

r

xy (n) is an estimate of the cross covarience

The cross-covarience is defined as

r xy

      1

N

1

N N k

  

n N k

 

n

 0  1

x

[ 0  1

x

[

k k

 ]

y

[

k n

] 

y

[

k

]

n

]

n

 0 , 1 , 2 ,...

n

 0 ,  1 ,  2 ,...

r xx

( 0 )  1

N N k

   1 0   2 ,

r yy

( 0 )  1

N N k

   1 0 

y

[

k

]  2

2.

with zero mean is given by 

xx

[

n

] 

xx

[

n

] 

r xx

[

n

]

r xx

[ 0 ] ,

n

 0 ,  1 ,  2

Digital Filtering

The equation for finite impulse response (FIR) filtering is

y

[

n

] 

k N

  1  0

h

[

k

]

x

[

n



k

] 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

y

x(n) 

N k

   0 1

b k



x



n



k

 z -1 x b 0 x b 1 z -1 x b 2 z -1 z -1 x b N-1 + +

Filter structure

+ A common filtering objective is to remove or reduce noise from a wanted signal. y(n)

(a) (b) (c) (d) (e) (f)

Figure

: Reconstructed bi-level text images for degradation caused by

h

1 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:

X

(

k

) 

N n

 1   0

x

[

n

]

W nk where W



e



j

2 

N

IDFT is defined as:

x

[

n

]  1

N k

 1

N

  0

X

(

k

)

W N



kn

, 0 

n



N

 1

MATLAB function for DFT

function [Xk] = dft (xn,N) % Computes Discrete Fourier Transform % ------------------------------------------------------ % Xk = DFT coefficient array over 0<= k <= N-1 % xn = N-point finite duration sequence % N = Length of DFT % n = [ 0:1:N-1]; k= [0:1:N-1]; WN = exp(-j*2*pi/N); nk = n.*k; WNnk = WN .^ nk; Xk = xn * WNnk;

Matlab Function for IDFT

function [xn] = idft(Xk,N) % Computes the Inverse Discrete Transform n = [ 0:1:N-1]; k= [0:1:N-1]; WN = exp(-j*2*pi/N); nk=n’*k; WNnk = WN .^(-nk); xn = (Xk * WNnk) / N;

Example

Let x[n] be a 4-point sequence

x

[

n

]  1 ,  0 , 0 

n

 3

otherwise

>>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.