# Basic Signal Operation

By Aina Parasher|Updated : October 3rd, 2022

Analyzing signals and systems certainly demands some basic signal operations. For the continuous-time signals, time and amplitude are two variables. Hence there may be a need to perform certain basic signal operations like amplitude-scaling, addition, subtraction, and multiplication with amplitude; and concerning time, the operations like time-scaling, time-shifting, and time-reversal.

Understanding certain basic signal operations can help make signals easier to understand and use in various situations. Check the operations on suitable signals below to better understand these basic signal operations.

## What is the Basic Signal Operation?

Signals can be subjected to several processes to produce new signals. Here, we have provided in-depth details about the following.

• Basic signal operations on Amplitude
• Basic signal operations on Time

## Basic Signal Operations on Amplitude

Concerning amplitude, we can perform four basic signal operations, namely-

• Amplitude Scaling
• Subtraction
• Multiplication

### Amplitude Scaling Signal Operation

The Amplitude scaling operation is applied to a continuous-time signal to increase or decrease its amplitude. If x(t) is any continuous-time signal, then.

Ax(t) is the amplitude scaling version of the signal x(t), where A is always a positive value.

If |A|>1→ Increase the amplitude of the signal

If |A|<1→ Decrease the amplitude of the signal

We can understand it better with the following example.

A = 1

|A|>1

|A|<1

If we add two continuous-time signals, x(t) and y(t), the resultant signal will have an amplitude equal to the sum of their amplitudes. The below example can be used to explain it in a better way.

From the above figures, the amplitude of the signal

z(t)=x(t)+y(t) between the interval -2<t<-1 is

z(t)=x(t)+y(t)=0+1=1

The amplitude of the signal z(t) between the interval −1<t<1 is

z(t)=x(t)+y(t)=1+1=2

The amplitude of the signal z(t) between the interval 1<t<2 isz(t) = x(t)+y(t) = 0+1 = 1.

### Subtraction Signal Operation

The subtraction operation of two signals is similar to that of addition, as the amplitude of the resultant signal is the value obtained from the subtraction of the amplitudes of the two individual signals in their respective intervals. Follow the below example to understand this better.

If z(t) is the signal that is obtained from subtracting y(t) from x(t) then the value of the amplitude of the signal z(t) in the interval

−2<t<−1 is

z(t)=x(t)−y(t)=10−0=10

The Amplitude of the signal z(t) in the interval −1<t<1 is

z(t)=x(t)−y(t)=10−5=5

The Amplitude of the signal z(t) in the interval 1<t<2 is

z(t)=x(t)−y(t)=10−0=10

### Multiplication Signal Operation

Just like addition and subtraction, the multiplication of two signals is the multiplication of the amplitudes of the two signals at their respective intervals. We can understand it better with the following example.

If z(t) is the signal that is obtained from multiplying the signals x(t) and y(t) then, the value of the amplitude of the signal z(t) in the interval

−2<t<−1 is

z(t)=x(t)×y(t)=2×0=0

The Amplitude of the signal z(t) in the interval −1<t<1 is

z(t)=x(t)×y(t)=2×1=2

The Amplitude of the signal z(t) in the interval 1<t<2

isz(t)=x(t)×y(t)=2×0=0

## Basic Signal Operations on Time

Concerning time, the following signal operations can be done.

• Time-scaling of Signals
• Time-shifting of Signals
• Time-reversal of Signals

### Time-Scaling of Signals

The time-scaling operation on the continuous-time signal is to expand or compress the signal on its time axis. If a continuous-time signal x(t) is scaled with a factor ‘a’ then

x(at)x(at) will represent its time-scaled version; here, ‘a’ is any positive value.

If |a|>1→a>1→Expansion of the signal

If |a|<1→a<1→Compression of the signal.

The below example will help us understand this concept better.

### Time-Shifting of Signals

The time-shifting of the continuous-time signal is shifting the signal to its left to make an advance or shifting to its right to induce some delay to the signal. If x(t) is a continuous-time signal, then x(t±t0) is the time-shifted version of x(t).

x(t+t0) → Advanced version (or) a Negative shift (or) Left shift

x(t−t0) → Delayed version (or) a Positive shift (or) Right shift

We can view the following example better by considering a unit step signal u(t).

### Time-Reversal of Signals

If we want the mirror-image of the signal across the vertical axis or to rotate vertically by an angle of 180º, then we operate time-reversal. If x(t) is a continuous-time signal, then x(-t) is its time-reversal version. Follow the below example for a better understanding.

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## FAQs on Basic Signal Operations

• Time and amplitude are two variable parameters for a continuous-time signal. We can perform the following operations on the amplitude of the signal- Amplitude scaling, Addition, Subtraction, and Multiplication.

With respect to time, we perform the operations like Time-scaling, Time-shifting, and Time-reversal.

• If x(t) is a continuous-time signal that is scaled with a factor ‘a’ on its time, then x(at) will be the resultant signal, here ‘a’ is always a positive value. If the value of ‘a’ is greater than 1, the resultant signal will expand in its time axis. On the other hand, if the value of ‘a’ is less than 1 then the resultant signal will be compressed in its time axis.

• If the time-shifting operation is applied on a continuous-time signal x(t), then the resultant signal is x(t±t0). If we want to induce some delay or, shift the signal towards its right then we perform the operation x(t-t0). If we need to advance the signal in the time axis or, shift the signal towards its left then we will perform the operation x(t+t0).

• If x(t) is a continuous-time signal with its amplitude scaled with a factor ‘A’ then’ Ax(t)' will be the amplitude scaled version of x(t). If the value of the scaling factor is greater than 1, then the amplitude of the signal will increase. If the value of the scaling factor is between 0 to 1 then the amplitude of the signal will decrease.

• The multiplication of two continuous-time signals is nothing but the multiplication of their amplitudes in their respective intervals. We can perform addition and subtraction in a similar way.

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