The signal operations can be performed on two variable parameters such as Amplitude and Time.
The signal operation with Amplitude involves :-
1. Amplitude Scaling
2. Addition
3. Subtraction
4. Multiplication
Amplitude Scaling :-
Let x(t) be the continuous time signal, then C x(t) is the amplitude scaled version of signal x(t) whose amplitude is scaled by the factor C.
Addition :-
Addition of two or more signals involves simply the addition of corresponding amplitude level of a signals.
-2 < t < -1 amplitude of y(t) = x1(t) + x2(t) = 1 + 0 = 1
-1 < t < 1 amplitude of y(t) = x1(t) + x2(t) = 1 + 2 = 3
1 < t < 2 amplitude of y(t) = x1(t) + x2(t) = 1 + 0 = 1
Subtraction :-
Subtraction of two or more signals involves simply the subtraction of corresponding amplitude level of a signals.
For y1(t):
-2 < t < -1 amplitude of y1(t) = x1(t) - x2(t) = 1 - 0 = 1
-1 < t < 1 amplitude of y1(t) = x1(t) - x2(t) = 1 - 2 = -1
1 < t < 2 amplitude of y1(t) = x1(t) - x2(t) = 1 - 0 = 1
For y2(t):
-2 < t < -1 amplitude of y2(t) = x2(t) - x1(t) = 0 - 1 = -1
-1 < t < 1 amplitude of y2(t) = x2(t) - x1(t) = 2 - 1 = 1
1 < t < 2 amplitude of y2(t) = x2(t) - x1(t) = 0 - 1 = -1
Multiplication :-
Multiplication of two or more signals involves simply the multiplication of corresponding amplitude level of a signals.
-2 < t < -1 amplitude of y(t) = x1(t) . x2(t) = 1 x 0 = 0
-1 < t < 1 amplitude of y(t) = x1(t) . x2(t) = 1 x 2 = 2
1 < t < 2 amplitude of y(t) = x1(t) . x2(t) = 1 x 0 = 0
The signal operation with Time involves :-
1. Time Scaling
2. Time Shift
3. Time Reversal
Time Scaling :-
Let x(t) be the continuous time signal, then x(At) is the time scaled version of signal x(t). Where A is always a positive.
|A| > 1 Compression of the signal
|A| < 1 Expansion of the signal
Time Shifting :-
Let x(t) be the continuous time signal, then x(t ± t0) is the time shifted version of the signal x(t).
x(t + t0) = left shift ; x(t - t0) = right shift
Time Reversal :-
Let x(t) be the continuous time signal, then x(-t) is the time reversal or reflection of the signal x(t).
Example Problems :
1. Plot the signal x(t) = -u(t+3) +2u(t+1) -2u(t-1) +u(t-3)
To find x(t), firstly we must sketch the signals,
x1(t) = -u(t+3), x2(t) = +2u(t+1), x3(t) = -2u(t-1), x4(t) = +u(t-3) as shown below.
The signal x(t) = x1(t) + x2(t) + x3(t) + x4(t) is obtained below and plotted in the above fig.
For t<-3; x(t) = x1(t) + x2(t) + x3(t) + x4(t) = 0+0+0+0 = 0;
For -3<t<-1; x1(t) = -1; x2(t) = x3(t) = x4(t) = 0; -1+0+0+0 = -1;
For -1<t<-1; x1(t) = -1; x2(t)=2; x3(t) = x4(t) = -1+2+0+0= 1;
For 1<t<3; x1(t) = -1; x2(t)=2; x3(t) = -2 x4(t) = 0; -1+2-2+0= -1;
For t>3; x1(t) = -1; x2(t)=2; x3(t) = -2; x4(t) = 1; -1+2-2+1 = 0;
2. For the continuous time signal x(t) shown in fig. below plot the signal y(t) = x(3t+2)
Firstly, we need to shift the signal to the left side with 2 points.
Now we have to compress the signal with a factor of 3 as shown below.
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