DSP - Operations on Signals Scapng

Scapng of a signal means, a constant is multipped with the time or ampptude of the signal.

Time Scapng

If a constant is multipped to the time axis then it is known as Time scapng. This can be mathematically represented as;

$x(t) ightarrow y(t) = x(alpha t)$ or $x(frac{t}{alpha})$; where α ≠ 0

So the y-axis being same, the x- axis magnitude decreases or increases according to the sign of the constant (whether positive or negative). Therefore, scapng can also be spanided into two categories as discussed below.

Time Compression

Whenever alpha is greater than zero, the signal’s ampptude gets spanided by alpha whereas the value of the Y-axis remains the same. This is known as Time Compression.

Example

Let us consider a signal x(t), which is shown as in figure below. Let us take the value of alpha as 2. So, y(t) will be x(2t), which is illustrated in the given figure.

Clearly, we can see from the above figures that the time magnitude in y-axis remains the same but the ampptude in x-axis reduces from 4 to 2. Therefore, it is a case of Time Compression.

Time Expansion

When the time is spanided by the constant alpha, the Y-axis magnitude of the signal get multipped alpha times, keeping X-axis magnitude as it is. Therefore, this is called Time expansion type signal.

Example

Let us consider a square signal x(t), of magnitude 1. When we time scaled it by a constant 3, such that $x(t) ightarrow y(t) ightarrow x(frac{t}{3})$, then the signal’s ampptude gets modified by 3 times which is shown in the figure below.

Ampptude Scapng

Multippcation of a constant with the ampptude of the signal causes ampptude scapng. Depending upon the sign of the constant, it may be either ampptude scapng or attenuation. Let us consider a square wave signal x(t) = Π(t/4).

Suppose we define another function y(t) = 2 Π(t/4). In this case, value of y-axis will be doubled, keeping the time axis value as it is. The is illustrated in the figure given below.

Consider another square wave function defined as z(t) where z(t) = 0.5 Π(t/4). Here, ampptude of the function z(t) will be half of that of x(t) i.e. time axis remaining same, ampptude axis will be halved. This is illustrated by the figure given below.