signalsystemslabmanual_fall2013

8/11/2019 SignalSystemsLabManual_Fall2013

1/72


2/72


3/72


4/72


5/72


6/72


7/72


8/72


9/72


10/72


11/72


12/72


13/72


14/72


15/72


16/72


17/72


18/72


19/72


20/72


21/72


22/72


23/72


24/72


25/72


26/72


27/72


28/72


29/72


30/72


31/72


32/72


33/72


34/72


35/72


36/72


37/72


38/72


39/72


40/72


41/72


42/72


43/72


44/72


45/72


46/72


47/72


48/72


49/72


50/72


51/72


52/72


53/72


54/72


55/72


56/72


57/72


58/72


59/72

Signals and Systems Lab

59

EXPERIMENT # 10

PROPERTIES OF FOURIER TRANSFORM (PART I)

LINEARITY:

The Fourier transform is linear; that is, if

and

then

Define x1(t)=rect(4

t ) as sum of shifted heaviside() functions and x2(t)= e-2tu(t) in MATLAB.

Apply linearity property on these two signals. Suppose

After you have derived Fourier transform of functions, definet for some fixed value e.g. t=[-5:0.01:5] and w=[-2:0.1:2], and rewrite the equations to plot the graphs.

syms t x1=heavi si de( t +2) - heavi si de( t - 2) ;

x2=exp( - t *2) *heavi si de( t ) ; x=2*x1+3*x2;

X1 =f our i er ( x1) ; X2 =f our i er ( x2) ;X =f our i er ( x) ;

t =[ - 5: 0. 01: 5] ; w=[ - 2*pi : . 1: 2*pi ] ; x1=heavi si de( t +2) - heavi si de( t - 2) ; x2=exp( - t *2) . *heavi si de( t ) ; x=2*x1+3*x2; X1 =2. / w. *si n( 2*w) ;

X2 =1. / ( 2+i *w) ;X =4. / w. *si n( 2*w) +3. / ( 2+i *w) ;

subpl ot ( 311) , pl ot ( t , x, ' l i newi dt h' , 2) , gr i d, t i t l e( ' pl ot of x( t ) =2*x_1 ( t ) +3*x_2 (t) ' subpl ot ( 312) , pl ot ( w, ( 2*X1+3*X2) , ' l i newi dt h' , 2) , gr i d, t i t l e( ' pl ot of2*X_1 ( w) + 3*X_2 ( w) ' subpl ot ( 313) , pl ot ( w, ( X) , ' l i newi dt h' , 2) , gr i d, t i t l e( ' pl ot of X( w) ' )


60/72


60

Also, write MATLAB code to compare the magnitude and phase spectra of the right side of theequation i.e. to the Fourier transform of the left side i.e.

.

The output should be as follows:


61/72


61

SHIFTING PROPERTIES:

TIME SHIFTING

The above equation shows that delaying a signal by to seconds does not change its amplitudespectrum. The phase spectrum, however, is changed by wto.

Prove the time shifting property for given x(t)

Output of your code should be as follows.

FREQUENCY SHIFTING

Prove the above Frequency shifting property. Given that


62/72


62

Output of your code should be as follows

Figure 1

Figure 2


63/72


63

Figure 3

-6 -4 -2 0 2 4 6-1

0

1

x(t) e (j2t)

-4 -3 -2 -1 0 1 2 3 40

2

4|F(x(t) e (j2t))|

-4 -3 -2 -1 0 1 2 3 40

2

4< F(x(t) e (j2t))


64/72


64

EXPERIMENT # 11

PROPERTIES OF FOURIER TRANSFORM (Part-II)

THE SCALING PROPERTY:

The function represents the function compressed by factor a. Similarly, a functionrepresents the function X(w) expanded in frequency by same factor a. The scaling

property states that time compression of a signal results in its spectral expansion, and timeexpansion of the signal results in its spectral compression.

TIME SCALING

Prove the Time Scaling property for given signal x1(t)

Output of your code should be as follows:

Figure 1:


65/72


65

Figure 2

Figure 3


66/72


66

DUALITIY:

Prove the Duality property. Given that

a)

where = 4 in our case

In the above equation we used the fact that because rect is an even

function. Output of your code should be as follows:


67/72


67

b)


-5 0 50

0.5

1

x(t)

-10 -5 0 5 100

0.2

0.4

0.6

0.8X(w)

-10 -5 0 5 100

0.2

0.4

0.6

0.8

x2(t)=1/(2+j*t)

-5 0 50

2

4

6

X2(W)

TIME DIFFERRNTIATION

Prove the Time differentiation property for given x(t)



68/72


68


69/72


69

EXPERIMENT # 12

Overview

1. Even & Odd Components of a Signal2. CONVOLUTION

EVEN & ODD Components of a Signal

E XAMPLE 1

a) Define the signal x(t) shown below in MATLAB.

Note: Since we require to flip the function in time domain and then add the two functions, so thetime duration defined for x(t) should span the complete duration required to plot x(t) and x(-t).For example, in this case x(t) is non zero for 0 t 2 and so x(-t) will be non zero for -2 t 0.So the time duration you define in MATLAB should at least be -2 t 2 as shown below to

properly execute the rest of operations.

b) Use the built-in MATLAB function fliplr() to define x(-t).

c) Use the formula to determine the even component of a given function i.e.in MATLAB.

d) Use the formula to determine odd component of a given function i.e.

in MATLAB.

e) Use subplot() function to plot the signals obtained in parts above.


70/72


70

E XAMPLE 2

Perform the same steps of determining and plotting even and odd components of a function asdone before for the signal f (t) shown below:

Observe that the given function f (t) is an even function so its odd component should be equal to zero f (-t) should look like f(t).

E XERCISE 1

Perform the same steps of determining and plotting even and odd components of a function forthe signal g(t) shown below:

Observe that the given function g(t) is an odd function so its even component should be equal tozero.Write your complete code in the space provided below:

Continuous Time Convolution

EXAMPLE 1

1. Define the functions x(t) and h(t) given below in MATLAB.


71/72


71

2. Use conv() function in MATLAB to convolve the two functions defined above.3. Plot x(t) , h(t) and convolution result y(t) in MATLAB on a single figure.4. Note that the result of convolution spans from (Tx1+Th1) t (Tx2+Th2), so plot the

graph of y (t) against this time.5. Your final result should look like the figure below:

EXERCISE 1 :

For the two functions x(t) and h(t) shown below1. Define the functions x(t) and h(t) in MATLAB.2. Use conv() function in MATLAB to convolve x(t) and h(t).3. Plot x, h and y on a single figure window using subplot() function.


72/72


EXERCISE 2:

For the two functions x(t) and h(t) shown below1. Define the functions x(t) and h(t) in MATLAB.2. Use conv() function in MATLAB to convolve x(t) and h(t).3. Plot x, h and y on a single figure window using subplot() function.

signalsystemslabmanual_fall2013

Documents