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EXPERIMENT- 1
AIM- WRITE A PROGRAM TO GENERATE FOLLOWING FUNCTIONS: UNIT
IMPULSE, UNIT RAMP, SINUSOIDAL, EXPONENTIAL, RANDOM SIGNAL.
APPARATUS REQUIRED: MATLAB
THEORY-
UNIT IMPULSE: IF A UNIT STEP FUNCTION U(T) IS DIFFERENTIATED WITH
RESPECT TO T, THE DERIVATIVE IS ZERO FOR TIME T GREATER THAN ZERO,
AND IS INFINITE FOR TIME T EQUAL TO ZERO,IMPULSE FUNCTION IS
DENOTED BY Δ (T). AND IT IS DEFINED AS Δ (T) = .AN IMPULSE OF
UNITY AMPLITUDE OCCURRING AT IT T = 0 GIVES THAT IT HAS AN AREA ‘Δ’
EQUAL TO UNITY. THE UNIT IMPULSE FUNCTION IS REPRESENTED AS
SHOWN IN FIG.
THE LAPLACE TRANSFORM OF THE UNIT IMPULSE FUNCTION IS
UNIT RAMP: IF THE UNIT STEP FUNCTION IS INTEGRATED WITH RESPECT TO
TIME T, THEN THE UNIT RAMP FUNCTION RESULTS. IT IS SYMBOLIZED BY
R(T). A UNIT RAMP FUNCTION INCREASES LINEARLY WITH TIME. A UNIT
RAMP FUNCTION MAY BE DEFINED MATHEMATICALLY AS
THE LAPLACE TRANSFORM OF THE UNIT RAMP FUNCTION IS
SINUSOIDAL: SINUSOIDAL SIGNALS ARE PERIODIC FUNCTIONS WHICH ARE
BASED ON THE SINE OR
COSINE FUNCTION FROM TRIGONOMETRY.
THE GENERAL FORM OF A SINUSOIDAL SIGNAL
WHERE COS (∙) REPRESENT THE COSINE FUNCTION
• WE CAN ALSO USE SIN(∙), THE SINE FUNCTION
– ΩOT +Φ OR 2ΠFOT +Φ IS ANGLE (IN RADIANS) OF THE COSINE
FUNCTION
• SINCE THE ANGLE DEPENDS ON TIME, IT MAKES X(T) A SIGNAL
– ΩO I S THE RADIAN FREQUENCY OF THE SINUSOIDAL SIGNAL
• FO IS CALLED THE CYCLICAL FREQUENCY OF THE SINUSOIDAL
SIGNAL
– Φ IS THE PHASE SHIFT OR PHASE ANGLE
– A IS THE AMPLITUDE OF THE SIGNAL
EXPONENTIAL SIGNAL:
MATLAB CODE:-
FOR UNIT-IMPULSE-
% Generation of unit impulse function t=-1:0.01:1; A=5; y=A*(t==0); subplot(2,2,1); plot(t,y); xlabel('time'); ylabel('amplitude'); title('unit impluse signal'); subplot(2,2,2); stem(t,y); xlabel('sequence'); ylabel('amplitude'); title('unit impluse sequence'); FOR UNIT-STEP- % Generation of unit step function t1=-10:0.01:10; A=5; y1=(t1>=0); subplot(2,2,3); plot(t1,y1); xlabel('time'); ylabel('amplitude'); title('unit step signal'); subplot(2,2,4); stem(t1,y1); xlabel('sequence'); ylabel('amplitude'); title('unit impluse sequence'); FOR RAMP SIGNAL - % Generation of Ramp Signal t=0:3:50; A=5;
y=At; subplot(2,2,1); plot(t,y); xlabel('time'); ylabel('amplitude'); title('Ramp Signal'); subplot(2,2,2); stem(t,y); xlabel('time'); ylabel('amplitude'); title('Ramp Signal'); SINC FUNCTION- % Generation of Sinc Signal t1=-5:0.01:5; y1=sinc(t1); subplot(2,2,3); plot(t1,y1); xlabel('time'); ylabel('amplitude'); title('Sinc Signal'); subplot(2,2,4); stem(t1,y1); xlabel('time'); ylabel('amplitude'); title('Sinc Signal'); FOR EXPONENTIAL SIGNAL- % Generation of Exponential Signal t=0:0.01:5; A=10; y=exp(A.t); y1=exp(-A.*t); subplot(2,2,1); plot(t,y); xlabel('Time'); ylabel('amplitude'); title('Growing Exponential signal'); subplot(2,2,2); stem(t,y); xlabel('Time'); ylabel('amplitude'); title('Growing Exponential sequence'); subplot(2,2,3); plot(t,y1); xlabel('Time'); ylabel('amplitude'); title('Growing Exponential signal'); subplot(2,2,4); stem(t,y1); xlabel('Time'); ylabel('amplitude'); title('Growing Exponential sequence');
RAMP SIGNAL:-
SINC FUNCTION:-
EXPONENTIAL SIGNAL:-
RANDOM SIGNAL:-
IT’S COMMAND WINDOW -
UNIT-STEP:-
CONCLUSION:- IN THIS EXPERIMENT WE LEARNED TO GENERATE,UNIT-
IMPULSE,UNIT-RAMP,SINUSOIDAL,EXPONENTIAL AND RANDOM SIGNAL
USING MATLAB.
EXPERIMENT- 2
AIM- WRITE A PROGRAM TO STUDY THE BASIC OPERATIONS ON THE
DISCRETE TIME SIGNALS: AMPLITUDE SCALING, TIME SHIFTING, TIME
SCALING, FOLDING, ADDITION AND MULTIPLICATION OF TWO SIGNALS.
APPARATUS REQUIRED: MATLAB
THEORY-
AMPLITUDE SCALING-
AMPLITUDE SCALING IS A VERY BASIC OPERATION PERFORMED ON
SIGNALS TO VARY ITS STRENGTH. IT CAN BE MATHEMATICALLY
REPRESENTED AS Y(T) = Α X(T).
HERE, Α IS THE SCALING FACTOR, WHERE: -
Α<1 → SIGNAL IS ATTENUATED.
Α>1 → SIGNAL IS AMPLIFIED.
THIS IS ILLUSTRATED IN THE DIAGRAM, WHERE THE SIGNAL IS
ATTENUATED WHEN Α = 0.5 IN FIG (B) AND AMPLIFIED WHEN Α = 1.5 AS IN FIG
(C)
TIME SHIFTING-
TIME SHIFTING IS, AS THE NAME SUGGESTS, THE SHIFTING OF A SIGNAL IN
TIME. THIS IS DONE BY ADDING OR SUBTRACTING AN INTEGER QUANTITY
OF THE SHIFT TO THE TIME VARIABLE IN THE FUNCTION. SUBTRACTING A
FIXED POSITIVE QUANTITY FROM THE TIME VARIABLE WILL SHIFT THE
SIGNAL TO THE RIGHT (DELAY) BY THE SUBTRACTED QUANTITY, WHILE
ADDING A FIXED POSITIVE AMOUNT TO THE TIME VARIABLE WILL SHIFT
THE SIGNAL TO THE LEFT (ADVANCE) BY THE ADDED QUANTITY.
TIME SCALING (CONTINUOUS TIME) -
A SIGNAL X(T) IS SCALED IN TIME BY MULTIPLYING THE TIME VARIABLE BY
A POSITIVE CONSTANT B, TO PRODUCE X(BT). A POSITIVE FACTOR OF B
EITHER EXPANDS (0 < B < 1) OR COMPRESSES (B > 1) THE SIGNAL IN TIME.
IN MULTIPLICATION:- LIKE ADDITION MULTIPLICATION OF SIGNALS ALSO
FALLS UNDER THE CATEGORY OF BASIC SIGNAL OPERATIONS. HERE
MULTIPLICATION OF AMPLITUDE OF TWO OR MORE SIGNALS AT EACH
INSTANCE OF TIME OR ANY OTHER INDEPENDENT VARIABLES IS DONE
WHICH ARE COMMON BETWEEN THE SIGNALS. THE RESULTANT SIGNAL WE
GET HAS VALUES EQUAL TO THE PRODUCT OF AMPLITUDE OF THE PARENT
SIGNALS FOR EACH INSTANCE OF TIME. MULTIPLICATION OF SIGNALS IS
ILLUSTRATED IN THE DIAGRAM BELOW, WHERE X 1( T) AND X 2( T) ARE TWO
TIME DEPENDENT SIGNALS, ON WHOM AFTER PERFORMING THE
MULTIPLICATION OPERATION WE GET,
Y(t)=X1(t).X2(t) TIME REVERSAL-
MATLAB CODE:-
SCALING OF THE SIGNAL-
%Scalingofthesignal t=-1:0.01:1; y1=sin(2pi4t); subplot(4,1,1); stem(t,y1); xlabel('Time'); ylabel('Amplitude'); title('InputSignal'); a=0.5; y2=sin(2pi4ta); subplot(4,1,2); stem(t,y2); xlabel('Time'); ylabel('Amplitude'); title('ExpandedSignal'); b=1.5; y3=sin(2pi4t*b); subplot(4,1,3); stem(t,y3); xlabel('Time'); ylabel('Amplitude'); title('CompressedSignal'); FOR TIME REVERSAL- % Time Reversal of the signal n=0:2:40; y=sin(n);
xlabel('time'); ylabel('amplitude'); title('signal-1'); subplot(2,2,2); stem(n,y2); xlabel('time'); ylabel('amplitude'); title('signal-2'); %Addition of signal-1 and signal- 2 y3=y1+y2; subplot(2,2,3); stem(n,y3); xlabel('time'); ylabel('amplitude'); title('addition'); %Multiplication of signal-1 and signal- 2 y4=y1*y2; subplot(2,2,4); stem(n,y4); xlabel('time'); ylabel('amplitude'); title('multiplication'); RESULTS- TIME SHIFTING:- ADDITION AND MULTIPLICATION OF TWO SIGNALS:-
TIME REVERSAL:-
CASUALITY- A CASUAL SYSTEM IS ONE WHOSE OUTPUT DEPENDS ONLY ON
THE PRESENT AND THE PAST INPUTS.A NON- CASUAL SYSTEM’S OUTPUT
DEPENDS ON THE FUTURE INPUTS.IN A SENSE, A NON-CASUAL SYSTEM IS
JUST THE OPPOSITE OF ONE THAT HAS MEMORY.
STABILITY- A SYSTEM IS SAID TO BE STABLE,IF ITS OUTPUT IS UNDER
CONTROL.OTHERWISE, IT IS SAID TO BE UNSTABLE.A STABLE SYSTEM
PRODUCES A BOUNDED OUTPUT FOR A GIVEN BOUNDED INPUT.
MATLAB CODE-
FOR LINEARITY:-
% Linearity of the system y(n)=x(n)^ x0=0; y0=x0^2; if y0== disp('System follows homogenity') %Additivity x1= x2=
x3=x1+x2; y1=x1^ y2=x2^ y3=x3^ z3=y1+y2; if y3==z disp('System is linear') else disp('System is non-linear') end end FOR CASUALITY:- % casuality of y(t)=x(t+2) t=input('enter the value of t:'); a=t+2; if a<=t disp('casual system'); else a>=t disp('non casual system') end FOR STABILITY:- %Stability of the system n=1:1:100; a=2; h=exp(-a.n); subplot(2,1,1); stem(n,h); for k=-10^2:1:10^ y=sum(exp(-a.k)); end b=-2; h=exp(-b.n); subplot(2,1,2); stem(n,h); for k=-10^2:1:10^ y=sum(exp(-a.k)) end RESULT:- LINEARITY:-
