Scaling - Operations on Signals | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

Scaling of a signal means, a constant is multiplied with the time or amplitude of the signal.

Time Scaling

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

Scaling - Operations on Signals | Signals and Systems - Electronics and Communication Engineering (ECE)

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, scaling can also be divided into two categories as discussed below.

Time Compression

Whenever alpha is greater than zero, the signal’s amplitude gets divided 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.

Scaling - Operations on Signals | Signals and Systems - Electronics and Communication Engineering (ECE)

 

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

Time Expansion

When the time is divided by the constant alpha, the Y-axis magnitude of the signal get multiplied 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 Scaling - Operations on Signals | Signals and Systems - Electronics and Communication Engineering (ECE)  then the signal’s amplitude gets modified by 3 times which is shown in the figure below.

Scaling - Operations on Signals | Signals and Systems - Electronics and Communication Engineering (ECE)

 

Amplitude Scaling

Multiplication of a constant with the amplitude of the signal causes amplitude scaling. Depending upon the sign of the constant, it may be either amplitude scaling 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.

Scaling - Operations on Signals | Signals and Systems - Electronics and Communication Engineering (ECE)

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

Scaling - Operations on Signals | Signals and Systems - Electronics and Communication Engineering (ECE)

The document Scaling - Operations on Signals | Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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FAQs on Scaling - Operations on Signals - Signals and Systems - Electronics and Communication Engineering (ECE)

1. What is scaling in the context of signal operations?
Ans. Scaling in signal operations refers to the process of multiplying a signal by a constant factor. It is a linear operation that affects the amplitude of the signal without changing its shape or duration. Scaling can either amplify or attenuate the signal depending on the scaling factor applied.
2. How does scaling affect the frequency content of a signal?
Ans. Scaling a signal does not affect its frequency content. When a signal is scaled, its frequency components remain unchanged. Scaling only alters the amplitude of the signal, not its frequency distribution. Therefore, the spectral characteristics of the signal, such as the peak frequencies and bandwidth, remain the same after scaling.
3. Can scaling a signal introduce distortion or artifacts?
Ans. Scaling a signal by a constant factor does not introduce any distortion or artifacts. It is a linear operation that simply multiplies the signal amplitude by the scaling factor. However, if the scaling factor is too high and exceeds the dynamic range of the system or equipment processing the signal, clipping or saturation may occur, leading to distortion. This is not a direct result of scaling but rather an issue of signal level management.
4. How does scaling affect the signal-to-noise ratio (SNR)?
Ans. Scaling a signal does not directly affect the signal-to-noise ratio (SNR). The SNR is determined by the ratio of the signal power to the noise power. Scaling the signal by a constant factor multiplies both the signal and noise components by the same factor, resulting in no change to their relative power levels. Hence, the SNR remains unchanged after scaling.
5. Can scaling be used to normalize the amplitude of different signals?
Ans. Yes, scaling can be used to normalize the amplitude of different signals. By applying scaling factors to individual signals, their amplitudes can be adjusted to a desired common level. This is particularly useful when combining multiple signals or comparing signals with different amplitudes. Normalizing the amplitudes allows for easier analysis and comparison, ensuring that the signals are on a similar scale.
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