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.

Addition

Addition of two signals is nothing but addition of their corresponding amplitudes. This can be best explained by using the following example:

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:

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:

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 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

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).