experiment8 sampling theory

8/11/2019 Experiment8 Sampling Theory

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Prepared By:


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Experiment 8:Sampling

ObjectiveThe objective of this Lab is to understand concepts and observe the effects of periodically sampling a

continuous signal at different sampling rates, changing the sampling rate of a sampled signal,aliasing, and anti-aliasing filters.

Introduction,A typical time-dependent signal, for example AC voltage, is continuous with respect to magnitude and

time. Such signals are called analog signals. Using a normal (analog) oscilloscope we get an analog

representation of such a signal.

Today mainly digital equipment is used for electrical measurements. The original analog signal is

converted to a digital signal. A digital signal is discrete with respect to the magnitude as well as to the

time. Therefore, conversion of an analog to a digital signal means the value of the analog signal

function F(t) is measured at discrete times

Sampling

In order to store, transmit or process analog signals using digital hardware, we must first convert them into

discrete-time signals by sampling.

The processed discrete-time signal is usually converted back to analog form by interpolation, resulting in a

reconstructed analog signalxr(t).

The sampler reads the values of the analog signalxa(t) at equally spaced sampling instants. The time interval

Tsbetween adjacent samples is known as the sampling period (or sampling interval). The sampling rate,measured in samples per second, isfs =1/Ts.

x[n]= xa(nTs) , n=, -1, 0, 1, 2, .

Also it possible to reconstructxa(t) from its samples: xa(t) =x[tFs].

Figure 4.1: Sampling and Reconstruction process


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Sampling Theorem

The uniform sampling theorem states that a bandlimited signal having no spectral components abovefm hetez can be determined uniquely by values sampled at uniform intervals of :

The upper limit on Tscan be expressed in terms of sampling rate, denoted fs=1/Ts. The restriction,

stated in term of the sampling rate, is known as the Nyquist criterion. The statement is :

The sampling rate is also called Nyquist rate.The allow Nyquist criterion is a theoreticallysufficient condition to allow an analog signal to be reconstructed completely from a set of a uniformly

spaced discrete-time samples.

Impulse Sampling

Assume an analog waveformx(t),as shown in figure 4.2(a), with a Fourier transform,X(f)which is

zero outside the interval (fm


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Using thefrequency convolution propertyof the Fourier transform we can transform the time-domain

product to the frequency-domain convolution ,

whereis the Fourier transform of the impulse train . Notice that the Fourier transform ofan impulse train is another impulse train; the values of the periods of the two trains are reciprocally

related to one another. Figure 4.2(c) and (d) illustrate the impulse train and its Fourier transform, respectively.We can solve for the transformof the sampled waveform:

*[ ]


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Figure 8.2: Sampling theorem using the frequency convolution property of the Fourier transform

We therefore conclude that within the original bandwidth, the spectrum is, to within a constantfactor (1/Ts) , exactly the same as that ofx(t) in addition, the spectrum repeats itself periodically in

frequency everyfshertz.


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Aliasing

Aliasing in Frequency Domain

If fsdoes not satisfy theNyquist rate, , the different components ofoverlap and willnot be able to recoverx(t)exactlyas shown in figure 8.3(b). This is referred to as aliasing in frequencydomain.

Figure 8.3: Spectra for various sampling rates. (a) Sampled spectrum (b) Sampled spectrum

:ampling theoryS

let

( ) : analoge signal

( ) ( ) : discrete signal

: analoge frequency Hz

: discrete frequency Hz

: analoge frequency rad/sec

: discrete frequency rad/sec

: sampling

( ) ( )

let the analoge signal

s s

s

s s

x t

x nT x n

F

w

T period

x n x nT

( ) cos(2 )

sampling ( ) cos(2 ) ( ) cos(2 )

Didital frequency (discrete frequency )=analoge frequency x sampling

s s s

s

t

nT

x t F

x nT F x n fn

period

f FT


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Figure 1shows a simple example .The solid linedescribes a 0.5Hz continuous-time sinusoidal signal and the

dash-dot line describes a 1.5 Hzcontinuous time sinusoidal signal. When both signals are sampled at the rate

of Fs =2 samples/sec, their samples coincide, as indicated by the circles. This means thatx1[nTs] is equal to

x2[nTs] and there is no way to distinguish the two signals apart from their sampled versions. This phenomenon

is known as aliasing.

Figure 1: Two sinusoidal signals are indistinguishable from their sampled versions

( )

(

)


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Practical Partsf0=1000;%Freqyency of sin

fs1=10000;%Sampling Frequency Fs>2Fm

fs2=1500;%Sampling Frequency Fs2Fm

Sampling With Fs


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Do you now that the human voice bandwidth lay between 0-4.5K Hz , Try different Fs in the

following code and see the effect of aliasing .

% Record your voice for 5 seconds.

fs=1000;

recObj = audiorecorder(fs,16,1);disp('Start speaking.')

recordblocking(recObj, 5);

disp('End of Recording.');

% Play back the recording.

play(recObj);

% Store data in double-precision array.

myRecording = getaudiodata(recObj);

% Plot the waveform.

plot(myRecording)


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HomeworkThe signal can be sampled at the rate to yield Aliasing will be observed for various a) Let =10 kHz and =1 kHz. plot x[n] usingstem.m.

b) Use subplot to plotx(t)for =300 Hz, 700 Hz, 1100 Hz and 1500 Hz. and =10 kHz

try using for loop

Comment on the result figure in (b):

c) Use subplot to plot x[n] for =10300 Hz, 10700 Hz, 11100 Hz and 11500 Hz..

Comment on the result figure in (c):

(1) The period:

(2) Aliasing


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Question 2:

Consider an analog signal x(t) consisting of three sinusoids

x(t)= cos(2t)+cos(8t)+cos(12t)

Using Matlab,(a)Show that if this signal is sampled at a rate of Fs = 5 Hz, it will be aliased with the

following signal, in the sense that their sample values will be the same:

xa(t)= 3cos(2t)

Part 2: Aliasing in Frequency Domain:

a) Construct the simulink model shown in Figure 8.6.b) use the attachment files of the spectrum analyzer and put it in you current diroctory

c)

Input a sine wave ofFo=5 Hz frequency from the signal generator1 and a sine wave ofFo=50 Hz frequency from the signal generator2. Choose a samplingtime of 0.001 sec for the pulse generator with pulse duration of 1% of the sampling

period. Choose the cut-off frequency of 120 for the Butterworth low-pass filter.

d) Start the simulation for 10sec and notice all visualizers(included them in your report).

e) Repeat steps 2 and 3 for a sine wave different sampling frequency Comment.

Figure 8.6: Simulink model to implement aliasing in frequency domain

Spectrum

Analyzer1

Signal

Generator2

Signal

Generator1

Scope4

Scope3Scope2

Scope1

Scope1

Pulse

Generator Product

Mux

butter

Analog

Filter Design

Add


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Why does a wheel seem to move backwards as it speeds up?

Try to answer in the light of aliasing theory

experiment8 sampling theory

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