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CHAPTER 4:DIGITAL PROCESSING OFCONTINUOUS-TIME

SIGNALS

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WHAT YOU WILL LEARN INTHIS LECTURE?

Concept of Continuous-time signal conversion todiscrete-time /digital signal or sequences i.e theSampling Theorem.

Aliasing arising from violating the SamplingTheorem.

Definitions of Nyquist rate, oversampling,undersampling and critical sampling.

The process of converting back to continuousanalog signal i.e. reconstruction andinterpolations.

Sampling of bandpass signal as oppose to

lowpass signal

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DIGITAL PROCESSING OFCONTINUOUS-TIME SIGNALS

STEP 1:- Convert CT signals to DT/digital signal orsequences.

STEP 2:- Computation or processing of thesesequences.

STEP 3:- The proceed sequences need to beconverted back to CT signal.

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INTRODUCTION

Most signals in real-life are Ct signals i.e.music, speech, and images.

Increasingly DT signal processing algorithms

used & implemented using DT analog @digital systems.

For processing CT signals by digital methods

:- need to use ADC and DAC.

So, it is necessary to determine relationsbetween the CT signal & its DT equivalent(both in time-domain & frequency domain).

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INTRODUCTION

Analog-to-Digital Converter (ADC): CT signalsinto digital form.

Digital-to-Analog Converter (DAC): Digitalsignals into CT signals.

Sample & Hold (S/H) circuit: Samples andholds the values for ADC.

Reconstruction (Smoothing) filter: Output the

DAC in staircase form. Helps smooth thesignal.

Anti-aliasing filter: Reduce the bandwidth ofthe CT signal to avoid aliasing due to

sampling.

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BLOCK DIAGRAM OF DSP

Block diagram representation

Idealized Block diagram representation

Anti-aliasingFilter

S/H ADCDigital

ProcessorDAC

Reconstructionfilter

IdealSampler

Discrete-

timeprocessor

IdealInterpolator

xa(t) x(n) y(n) ya(t)

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CONTINUOUS TO DIGITAL SIGNALCONVERSION

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IMPULSE-TRAIN SAMPLING

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MULTIPLICATION / MODULATIONPROPERTY

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Since is treated as periodic signal with period T.

From Fourier series coefficients;

and

MULTIPLICATION / MODULATIONPROPERTY

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Integration exit only when

MULTIPLICATION / MODULATIONPROPERTY

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CONVOLUTION IN FREQUENCYDOMAIN

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SAMPLING THEOREM

x(t) is sampled every T sec X[nT] wheren = 0,1, 2,

x(t) is band-limited i.e. X(j)=0 for ||> M

x(t) can be uniquely recovered if we sample atthe rate:-

s > 2M (Nyquist rate) where s=2/T.

i.e. Sampling period; T< /M

T sec

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RELATIONSHIP BETWEEN ANALOGFREQUENCY() & DIGITAL FREQUENCY()

?

Using the time-shifting property, we note that CTFT of :

ICTFT

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VIOLATING SAMPLING THEOREMRESULTING IN ALIASING

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EFFECT OF SAMPLING IN THEFREQUENCY DOMAIN

If sampling frequency higher than Nyquist rate;

Sampling operation referred to as oversampling.

If sampling frequency lower than Nyquist rate;

Sampling operation referred to as undersampling.

If sampling frequency equal to Nyquist rate;

Sampling operation referred to as critical sampling.

T > 2M

T < 2M

T = 2M

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EXAMPLE 1

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A digital telephony sampling rate used is8KHz. Is 3.4KHz signal bandwidth issufficient for telephone conversation?Answer:

YES

HOW???FT = 8KHz, FM = 3.4KHz.

Therefore; 2FM = 6.8KHz

FT > 2FM

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EXAMPLE 2

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SAMPLING IN HIGH QUALITYMUSIC SYSTEMS

For high quality analog music signal processing,a 20KHz signal bandwidth determine the highfidelity.

Therefore, in compact disc (CD) digital musicsystem, a sampling frequency (rate) of44.1KHz

which is greater than twice the signalbandwidth, is used.

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EXAMPLE 3

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EXAMPLE OF ALIASINGILLUSTRATION

Consider three continuous-time sinusoidal signals:-

x1(t)=cos(6t), x2(t)=cos(14t) and x3(t)=cos(26t)

(a) Determine the CTFT of the signals and sketch thespectrum of above transforms.

(b) At sampling rate T=0.1s, determine each cosinesignal condition.

(c) Sketch the spectrum of each sampled cosinesignal.

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ALIASING

Aliasing to OCCUR;

F : FT < 2FM : T < 2M

Aliasing NOT OCCUR;

F : FT > 2FM

: T > 2M