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# Signals and Systems: Amplitude Scaling of Signals

## What is Amplitude Scaling?

The process of rescaling the amplitude of a signal, i.e., the amplitude of the signal is either amplified or attenuated, is known as **amplitude scaling**. In the amplitude scaling operation on signals, the shape of the resulting signal remains
the as that of the original signal but the amplitude is altered (i.e., increased or decreased).

## Amplitude Scaling of a Continuous-Time Signal

The amplitude scaling of a continuous time signal x(t) is defined as,

𝑦(𝑡) = 𝐴 𝑥(𝑡)

Where, A is a constant. If the value of A is greater than 1 (i.e., A > 1), the signal amplitude scaling is called the amplification of the signal while if A < 1, then the scaling is called the **attenuation of the signal**.

Figure-1 shows an arbitrary continuous-time signal x(t) and its amplitude scaled version y(t).

## Amplitude Scaling of a Discrete-Time Signal

The amplitude scaling of a discrete time sequence x(n) is defined as,

𝑦(𝑛) = 𝑘 𝑥(𝑛)

Where, k is a constant. If k > 1, the scaling is called amplification of the signal, while if k < 1, the scaling is called **attenuation of the signal**.

An arbitrary discrete time sequence x(n) and its scaled version y(n) are shown in Figure-2.

## Numerical Example

Sketch the following continuous-time signal x(t) −

𝑥(𝑡) = Π(𝑡); 𝑦(𝑡) = 2 Π(𝑡)

𝑥(𝑡) = 𝑢(𝑡); 𝑦(𝑡) = −2𝑢(𝑡)

### Solution

The given signal is a unit rect function, i.e.,

𝑥(𝑡) = Π(𝑡)

The amplitude scaled signal [𝑦(𝑡) = 2 Π(𝑡)] of the given signal is obtained by multiplying the amplitude of x(t) by 2 as shown in Figure-3. Here, it is to be noted that the value of scaling factor more than 1 (i.e., 2), hence the resulting signal will be an amplified signal.

The given signal is a unit step function, i.e.,

𝑥(𝑡) = 𝑢(𝑡)

Here, in order to obtain the scaled signal [𝑦(𝑡) = −2𝑢(𝑡)], the amplitude of the signal x(t) needs to be multiplied by (-2). Figure-4 shows the signals x(t) and y(t).

- Related Articles
- Signals and Systems – Time Scaling of Signals
- Signals and Systems – Classification of Signals
- Signals and Systems: Multiplication of Signals
- Signals and Systems: Addition and Subtraction of Signals
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- Signals and Systems – Fourier Transform of Periodic Signals
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