Signals Basic Operations

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Operations of CT Signals
                   1. Time Reversal y(t) = x(-t)
                   2. Time Shifting y(t) = x(t-t d )
                   3. Amplitude Scaling y(t) = Bx(t)
                   4. Addition y(t) = x 1(t) + x 2(t)
                   5. Multiplication y(t) = x 1(t)x 2(t)
                   6. Time Scaling y(t) = x(at)

There are two variable parameters in general:
  1. Amplitude
  2. Time
The following operation can be performed with amplitude:

Amplitude Scaling

C x(t) is a amplitude scaled version of x(t) whose amplitude is scaled by a factor C.
Amplitude scaling

Addition

Addition of two signals is nothing but addition of their corresponding amplitudes. This can be best explained by using the following example:
Amplitude addition
As seen from the diagram above,
    -10 < t < -3 amplitude of z(t) = x1(t) + x2(t) = 0 + 2 = 2
    -3 < t < 3 amplitude of z(t) = x1(t) + x2(t) = 1 + 2 = 3
    3 < t < 10 amplitude of z(t) = x1(t) + x2(t) = 0 + 2 = 2

Subtraction

subtraction of two signals is nothing but subtraction of their corresponding amplitudes. This can be best explained by the following example:
Amplitude subtraction
As seen from the diagram above,
    -10 < t < -3 amplitude of z (t) = x1(t) - x2(t) = 0 - 2 = -2
    -3 < t < 3 amplitude of z (t) = x1(t) - x2(t) = 1 - 2 = -1
    3 < t < 10 amplitude of z (t) = x1(t) + x2(t) = 0 - 2 = -2

Multiplication

Multiplication of two signals is nothing but multiplication of their corresponding amplitudes. This can be best explained by the following example:
Amplitude multiplication
As seen from the diagram above,
    -10 < t < -3 amplitude of z (t) = x1(t) ×x2(t) = 0 ×2 = 0
    -3 < t < 3 amplitude of z (t) = x1(t) ×x2(t) = 1 ×2 = 2
    3 < t < 10 amplitude of z (t) = x1(t) × x2(t) = 0 × 2 = 0
The following operations can be performed with time:

Time Shifting

x(t ± t0) is time shifted version of the signal x(t).
    x (t + t0 negative shift
    x (t - t0 positive shift
Time shifting

Time Scaling

x(At) is time scaled version of the signal x(t). where A is always positive.
    |A| > 1  Compression of the signal
    |A| < 1  Expansion of the signal
Time scaling
Note: u(at) = u(t) time scaling is not applicable for unit step function.

Time Reversal

x(-t) is the time reversal of the signal x(t).
Time reversal

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