Report of Matlab

8/7/2019 Report of Matlab

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[DIGITAL COMMUNICATION SIMULATION USING MATLAB] December 7, 2010

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INTRODUCTION :

AIM: To model and analyse the Communication System Using MATLAB

OBJECTIVE:

1) Study ofthe Matlab basics

2) Study offft andfftshift commands

3) Performing time-continuous signal analysis

4) Performing spectrum analysis using the FFTfunctions

SCOPE:

1) MATLAB has a wide scope in Wireless Communication Systems

2) In Signal and Image processing, communications, control design, test and

measurement, financial modelling and analysis, and computational biology

Communication Systems:

Communication Systems can be defined as the systems which are designed to

transmit information. Basically there are two types of communication systems namely

Analog Communication Systems and Digital Communication Systems. An Analog

Communication System is used to transmit the analog information while the Digital

Communication System is used to transmit the digital information. Regardless of any

particular information any communication system consists of three main subsystems

namely transmitter, channel and receiver. (W.Couch, fifth edition)

Fig 1: Basic Communication System (W.Couch, fifth edition)


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The [~] indicates that the received signal is not the same as the transmitted signal.

In analog communication system the Signal Processing block (processor) may be an

analog low-pass filter which is used to limit the band-width of m(t) which is the input

information signal. In digital communication system the Signal processor may be Analog to

Digital Converter (ADC) which is used to convert the analog signals intodigital signals. In

addition the Signal Processor also provides source coding of the input signals. It also adds

parity bits to provide the channel coding by which error detection and correction can be

made possible at the receiver to eliminate the bit errors introduced by the noise in the

channel. The transmitter signal processor output signal is the base-band signal because it

has the frequencies concentrated around the f=0. (W.Couch, fifth edition)

The transmitter carrier circuit c

onverts the pr

ocesse

dbaseban

dsignal int

oa

frequency band that is appropriate for the transmission medium of the channel. Carrier

circuits are needed when the transmission channel is located in a band of frequencies

around fc where fc >> 0. fc is the carrier frequency. Here s(t) is the band pass signal.

(W.Couch, fifth edition)

Channels can be categorized into two types, hard wire and soft wire. Examples of

hard wire channels are coaxial-cables, wave guides, twisted pair telephone lines, fiber

optical cables etc. Examples of soft wire channels are air, vacuum and sea-water. The

channels contain active amplifying devices, such as repeaters in telephone line systems or

satellite transponders in space communication systems which helps to keep the signal

above the noise level. (W.Couch, fifth edition)

The receiver receives the corrupted signal from the channel output and converts into

baseband signal which can be handled by the receiver base band processor. This signal is

cleaned by the base band processor anddelivers the estimate of the source information to

the communication system output. (W.Couch, fifth edition)

The communication systems which use atmosphere as the transmission channel,

interference and propagation conditions are strongly dependent on the transmission

frequency. (W.Couch, fifth edition)


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Spectrum Analysis:

Examination of signals in the frequency domain is one of the most important

measurement tasks in radio communications. The most versatile and widely used RF

measuring instruments required for this purpose are Spectrum analyzers. Covering

frequency ranges ofup to 40 GHz and beyond, they are used in practically all applications of

wireless and wired communication in development, production, installation and

maintenance efforts. With the growth of mobile communications, parameters such as

displayed average noise level, dynamic range and frequency range, and other exacting

requirements regarding functionality and measurement speed come to the fore. Moreover,

spectrum analyzers are also usedfor measurements in the time domain, such as measuring

the transmitter output power of time multiplex systems as a function of time. (Rauscher,

2008)

Spectrum analysis is the process of determining the frequency domain

representation ofa time domain signal and most commonly employs the Fourier transform.

The general frame of reference is Time. In the time domain the amplitude of

electrical signals is plotted versus time. An oscilloscope is used to see the instantaneous

value of a particular electrical event (or some other event converted to volts through an

appropriate transducer) as a function oftime. (unknown, Agilent Spectrum Analysis Basics)

According to the Fourier theorem, any signal that is periodic in the time domain can

be derived from the sum of sine and cosine signals ofdifferent frequency and amplitude.

Such a sum is referred to as a Fourier series. (Rauscher, 2008)

(Rauscher, 2008)

With proper filtering any waveform can be decomposed into separate sine waves, or

spectral components, that can be evaluated independently later.A Spectrum can be defined

as the collection of sine waves that, when combined properly, produce the time-domain

signal. (unknown, Agilent Spectrum Analysis Basics)

The frequency domain has its measurement strengths. The frequency domain is

better for determining the harmonic content of a signal. Communications people are


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generally more interested in harmonic distortion. Consider an example, where cellular radio

systems are checked for harmonics of the carrier signal which might interfere with other

system operating at the same frequencies as the harmonics. Communications people are

also interested in distortion of the message modulated onto a carrier. Third-order inter

modulation (two tones of a complex signal modulating each other) can be particularly

troublesome because the distortion components can fall within the bandof interest and so

not be filtered away. (unknown, Agilent Spectrum Analysis Basics)

A practical quantitative assessment of the higher-order harmonics is not possible in

the Time domain analysis ofa signal. In the frequency domain it is much easier to examine the

short-term stability offrequency and amplitude of a sinusoidal signal compared to the time

domain. (Rauscher, 2008)

FFT:

FFT is abbreviated as Fast Fourier Transform. It is considered to be an efficient

algorithm to compute the Discrete Fourier Transform (DFT) and its inverse.

A DFTdecomposes a sequence of values into components ofdifferent frequencies.

This operation is useful in many fields, but computing it directly from the definition is often

too slow to be practical. An FFT is a way to compute the same result more quickly:

computing a DFTofN points in the simple way, using the definition, takes O(N2) arithmetical

operations, while an FFT can compute the same result in only O(N log N) operations. The

difference in speed can be substantial, especially for long data sets where N may be in the

thousands or millionsin practice, the computation time can be reduced by several orders

ofmagnitude in such cases, and the improvement is roughly proportional toN / log(N). This

huge improvement made many DFT-based algorithms practical; FFTs are of great

importance to a wide variety of applications, from digital signal processing and solving

partiald

iff

erential equatio

ns to

algo

rithmsfo

r quick multiplicatio

nof

large integers.(unknown, Fast Fourier Transform)

The DFT is extremely important in the area of frequency (spectrum) analysis

because it takes a discrete signal in the time domain and transforms that signal into its

discrete frequency domain representation. Without a discrete-time to discrete-frequency


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transform we would not be able to compute the Fourier transform with a microprocessor or

DSP based system. (unknown, FFTTutorial)

The DFT is NOT the same as the DTFT (Discrete Time Fourier Transform). Both start

with a discrete-time signal, but the DFT produces a discrete frequency domain

representation while the DTFT is continuous in the frequency domain. These two transforms

have much in common, however. (unknown, FFTTutorial)

The FFTdoes not directly give the spectrum of a signal. The FFT varies dramatically

depending on the number ofpoints (N) of the FFT, and the number ofperiods of the signal

that are represented. There is another problem as well. The FFT contains information

between 0 and fs, however, the sampling frequency must be at least twice the highest

frequency c

omp

onent.

There

fore, the signal's spectrum sh

ould

be entirely belo

wfs/2 , the

Nyquist frequency. (unknown, FFTTutorial)

The FFToperates by decomposing an N point time domain signal intoN time domain

signals each composed of a single point. The second step is to calculate the N frequency

spectra corresponding to these N time domain signals. Lastly, the N spectra are synthesized

into a single frequency spectrum. (unknown, Fast Fourier Transform)

This method is efficient because it reduces the redundancies which result from

adding the certain data sequence values after they have been multiplied by the same factors

of fixed complex constants during the evaluation ofdifferent DFT transform coefficients.

(F.Elliot, 1987)

Cooley-Tukey algorithm is one of the best algorithm used to compute the N point

DFT when N is composite (i.e. when N1N2). In this case it first computes the N1 transforms of

size N2 and then N2 transforms of size N1.

The

fft c

omman

din Matlab is use

dto

evaluate DT

FS and

find

the time-band

wid

thproduct property for discrete-time periodic signals. Since DFTS applies to signals that are

periodic both in time andfrequency, both the duration and bandwidth for the signal within

one period are to be defined. (Veen, 1999)


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Ifwe compute the Fourier transform of a sine or a cosine function we expect to see

two impulses at the frequency of the sine or the cosine. But, they cannot be seen. This can

be achieved by using fftshift function. (El-Osery, October 27, 2004)

FFTSHI

FT:

FFTSHIFT rearranges the fft output, moving the zerofrequency to the center of the

spectrum. The sequence can be given as y=fftshift(x [,job])

The parameters, x: real or complex vector or matrix

y: real or complex vector or matrix

job: integer, dimension selection, or string all

1) If x results of an fft computation y= fftshift(x) or y= fftshift(x,"all") moves the zero

frequency component to the center of the spectrum, which is sometimes a more

convenient form.

2) Ifx is a vector of size n, y is the vector x([n/2+1:n,1:n/2])

3) Ifx is an m by n matrix y is the matrix x([m/2+1:n,1:m/2],[n/2+1:n,1:n/2]).

y= fftshift(x,n) make the swap only along the nth dimension. (unknown, Scilab

Reference Manual)

In the formula y=fftshift(x)

Input x is a two-dimensional array or a vector, typically the output offft offft2 .

Output y is an array of the same size as the input. The output contains the same elements

but in a different order.

The description is given as

For a one-dimensional array, the element x[k+1] of the output x of fft is the

transform of the input at the frequency exp(2*pi*i*k/N), k=0,1,...N-1 where N is the size ofx

. Likewise, the element z[j+1,k+1]of the output z offft2 is the transform of the input at the

frequencies exp(2*pi*i*j/M), exp(2*pi*i*k/N) with j=0,1,...M-1 , and k=0,1,...N-1 . Here M

and N are the row and column dimensions ofy.


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fftshift rearranges the outputs of fft and fft2 so that the zero frequency is at the

center of the spectrum. If the input is a vector, fftshift swaps the upper and lower halfofthe

vector. If the input is a matrix, the first and third quadrants as well as the second andfourth

quadrants are swapped. (unknown, Help for fftshift)

Example: (unknown, Help for fftshift)

>>y=rand(1024,1);

>>z=fft(y)

>>subplot(2,1,1)

>>plot(abs(z))

>>title("FFT")

>>w=fftshift(z)

>>subplot(2,1,2)

>>plot(abs(w))

>>title("FFTHIFT")

(unknown, Help for fftshift)

Figure 2: fft andfftshift

It is helpful in visualizing a Fourier transform with the zero-frequency component in

the middle ofthe spectrum.

Consider an another example (El-Osery, October 27, 2004)


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>> fs=100;

>> t=0:1/fs:1;

>> y=cos(2*pi*t);

>>Y=fft(y);

>>plot(fftshift(abs(Y))) % gives the Figure 3

Now to map the x-axis value to reflect frequency in Hz, i.e. to get the positive and negative

frequencies

>> k=-N/2:N/2-1;

>> plot(k,fftshift(abs(Y))) % gives the Figure 4

Now to map it to actual frequencies

>> plot(k*fs/N,fftshift(abs(Y))) % gives the Figure 5

(El-Osery, October 27, 2004)

Figure 3: fftshift signal


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Figure 4: fftshift with k= -N/2 : N/2-1


Figure 5: Frequency mapping


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Task 1: FFT of a Cosine when N=n & N


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title('when N


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The second subplot shows the FFTofa Cosine Signal when the number of samples in

the signal is greater than the number of points in the FFT. When the number of points in

the FFT is less than the samples of the signal then the amplitude ofFFT signal gets lowered

which can be clearly observed in the above figure second subplot. This also results in the

irregular shaped two triangular waveforms with the center of the first waveform at 0.1fs and

the second at 0.9fs.

Task 2: FFT of a Cosine Signal with 3, 6, 9 & 12 Periods

In the previous task the number ofperiods is considered as 3 with the number ofFFT

points as N=30 and N=25. Now in this task taking N=2048 the number of repetitions of the

fundamental period is varied. 3 periods, 6 periods, 9 periods and 12 periods of a Cosine

Signal in length are considered.

Code Program:

n=[0:35];

X1=cos(2*pi*n/12);

X2=[X1 X1];

X3=[X1 X1 X1];

X4=[X1 X1 X1 X1];

N=2048;

Y1=abs(fft(X1,N));

Y2=abs(fft(X2,N));

Y3=abs(fft(X3,N));

Y4=abs(fft(X4,N))

F=[0:N-1]/N;


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subplot(2,2,1)

plot(F,Y1,'-x')

title('for 3 periods')

subplot(2,2,2)

plot(F,Y2,'-x')


subplot(2,2,3)

plot(F,Y3,'-x')


subplot(2,2,4)

plot(F,Y4,'-x')


This code results in the following figure 7

Figure 7: FFTofa Cosine with 3, 6, 9 and 12 periods


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The code results in the 4 subplots as shown in the above figure. The first

subplot shows the transform of 3 periods of a Cosine signal, and it looks like the

magnitude of2 Sincs with the center of the first Sinc at 0.1fs and the second at 0.9fs.

The second sub plot also looks like a Sinc, but with a higher frequency and with a

larger magnitude at 0.1fs and 0.9fs. Similarly, the third and the fourth subplots have

the larger Sinc frequencies and magnitudes. This clearly shows that as x[n] is

extended to the large number ofperiods, the Sincs will start looking more and more

like impulses. (unknown, FFTTutorial)

Task 3: Spectrum of an FFT signal using fftshift

Ifwe compute the Fourier transform ofa sine or a cosine function we expect to see

two impulses at the frequency ofthe sine or the cosine. But, that's not what we are seeing.(El-Osery, October 27, 2004)

The FFTdoesnt give directly the spectrum of a signal. The last two experiments

clearly explain that, the FFT varies dramatically with the number ofpoints (N) ofthe FFT,

and the number ofperiods ofthe signal that are represented. There is another problem as

well.

The FFT contains information between 0 andfs , however, it is known that the

sampling frequency must be at least twice the highest frequency component. Therefore, the

signal's spectrum should be entirely below fs/2, the Nyquist frequency.

Also a real signal should have a transform magnitude that is Symmetrical for positive

and negative frequencies. So insteadofhaving a spectrum that goes from 0 tofs, it would

be more appropriate to show the spectrum from fs/2 tofs/2 . This can be achieved by

using Matlab's fftshift function as the following code demonstrates. (unknown, FFT

Tutorial)

Code Program:

n=[0:209];

X=cos(2*pi*n/10)+sin(3*pi*n/10);

N=2048;

X1=abs(fft(X,N));

X2=fftshift(X1);


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F= [-N/2:N/2-1]/N;

plot(F,X2,'-x')

title('fftshift signal')

xlabel('frequency')

ylabel('amplitude')

This Code results the following figure 8

Figure 8: fftshift ofa Sum ofCosine Signal and Sine Signal

In the above figure the spectrum of the FFTof the Sum of the Cosine and Sine Signal

is spread around the center 0 which can be observed clearly in the above figure. The

spectrum looks like a symmetrical signal around the zero frequency. There are four peaks

twoon either side of the zerofrequency and it almost looks like an impulse signal. With the

increase of the length ofthe window, the sinc function almo st looks like an impulse signal.


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Conclusion:

By observing all the three above tasks it can be concluded saying that Matlab is a

powerful numerical computing environment and programming tool. Matlab is used in real

time applications like in wireless digital communication systems todesign and analyse the

spectrum of the different FFT signals. The fft command in the Matlab is considered to be the

powerful function to perform the DFTof the signal. The analysis in the frequency domain is

very useful and provides the clear understanding of the signal compared to the time

domain analysis.

It is observed that the change in the number of points (N) in the FFT, number of

samples of the signal and the number of repetitions of the fundamental period affect the

shape of the spectrum of the signal. The basic required condition is that, the number of

points in the FFT should be equal or greater than the number ofsamples in the signal.

It is also studied with the increase in the number ofperiods the sinc pulse turns into

nearly to the shape ofthe impulse which is the required criteria.

fftshift command in the Matlab is used to shift zero-frequency component to center

of spectrum. It is studied in the task 3 that the fftshift shifts the spectrum with 0 as the

center ofthe spectrum looking like symmetric signals on either side.


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References:

El-Osery, A. (October 27, 2004). Fast Fourier Transform. MATLAB Tutorial, 18.

F.Elliot, D. (1987). Handbook of Digital Signal Processing Enginnering Applications.California:

Acadaemic Press.

Rauscher, C. (2008). Fundamentals of Spectrum Analysis. Germany: Rohde & Schwarz GmbH & Co.

KG.

unknown. (n.d.). Agilent Spectrum Analysis Basics. Retrieveddecember 01, 2010, from

www.google.com.

unknown. (n.d.). Fast Fourier Transform. Retrieveddecember 1st, 2010, from www.wikipedia.com:

http://en.wikipedia.org/wiki/Fast_Fourier_transform

unknown. (n.d.). Fast Fourier Transform. Retrieveddecember 1st, 2010, from www.dspguide.com:

http://www.dspguide.com/ch12/2.htm

unknown. (n.d.). FFTTutorial. ELE 436: Communication Systems , 6.

unknown. (n.d.). Help for fftshift. Retrieveddecember 1st, 2010, from www.google.com:

http://www.mathnium.com/help/fftshift.html

unknown. (n.d.). Scilab Reference Manual. Retrieveddecember 1st, 2010, from www.google.com:

http://cermics.enpc.fr/~jpc/mopsi/doc/fftshift.html

Veen, S. H. (1999). Signals and Systems. Newyork: John Wiley & Sons.

W.Couch, L. (fifth edition). Digital and Analog Communication Systems. USA: Prentice Hall.


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Bibliography

F.Elliot, D. (1987). Handbook of Digital Signal Processing Enginnering Applications.California:

Acadaemic Press.

Rauscher, C. (2008). Fundamentals of Spectrum Analysis. Germany: Rohde & Schwarz GmbH & Co.

KG.

Veen, S. H. (1999). Signals and Systems. Newyork: John Wiley & Sons.

W.Couch, L. (fifth edition). Digital and Analog Communication Systems. USA: Prentice Hall.

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