When we talk about different types of operations performed on a signal, the concepts of shifting, scaling, and reversal come out. They are the basic operations that must be understood for signal sketching or simply drawing a signal. These operations not only help you sketch, but they help you understand the behavior of signals under different operations. But sometimes, they are a little bit confusing and it is difficult for us to differentiate between these terms. In this article, we will discuss these three terms in detail and we will also discuss some basic differences between them, that will help you understand better.
What is a Signal?
A signal is defined as the source of information that is converted into electrical form. But talking about a more technical definition, it is defined as the function of one or more than one independent variable.
The general form of a signal is: f(x1 , x2 , x3 , … xn )
Depending on the number of variables, there are two different types of signals, single and multivariable signals.
Types of Signal
There are two different types of signals: 1) Continuous-time signal and 2) Discrete-time signal.
- Continuous-time signal (CTS): It is a signal that is specified for all values of time. It is denoted by x(t).
- Discrete-time signal(DTS): It is a signal that is specified only for a certain value of time. It is denoted by x[n].
Different Operations Performed on the Signal
There are different types of operations performed on the signals as follows:
- Scaling
- Shifting
- Reversal
- Differentiation
- Integration
- Convolution
What is Time Scaling?
It is defined as the compression or expansion of signal in time. In time scaling we change the speed of signal flow. The general expression of time scaling is:
Time Scaling: x(t) → y(t) = x(αt), α ≠0 where α is a constant
Depending on the value of | α |, there are two different cases for time scaling.
Case 1: When | α | > 1, it is the case of Compression.
Let a signal is defined as: x(t) = 0 , at t < 0
= 2 , at 0 <= t <= 2
= 0 , at t > 0 In this article, all the examples are considered with respect to this signal
1, it is the case of Compression” width=”1024″ height=”inherit” srcset=”https://media.geeksforgeeks.org/wp-content/uploads/20230815003518/imgonline-com-ua-resize-eOfC310wyzSq.jpg 1100w,https://media.geeksforgeeks.org/wp-content/uploads/20230815003518/imgonline-com-ua-resize-eOfC310wyzSq-100.jpg 100w,https://media.geeksforgeeks.org/wp-content/uploads/20230815003518/imgonline-com-ua-resize-eOfC310wyzSq-200.jpg 200w,https://media.geeksforgeeks.org/wp-content/uploads/20230815003518/imgonline-com-ua-resize-eOfC310wyzSq-300.jpg 300w,https://media.geeksforgeeks.org/wp-content/uploads/20230815003518/imgonline-com-ua-resize-eOfC310wyzSq-660.jpg 660w,https://media.geeksforgeeks.org/wp-content/uploads/20230815003518/imgonline-com-ua-resize-eOfC310wyzSq-768.jpg 768w,https://media.geeksforgeeks.org/wp-content/uploads/20230815003518/imgonline-com-ua-resize-eOfC310wyzSq-1024.jpg 1024w”>When | α | > 1, it is the case of Compression
Case 2: When | α | < 1, it is the case of Expansion.
In time scaling, the amplitude of the signal remains the same, only time is changing according to the value of α. The compression and expansion lead to a change in the shape of the original signal.
<img src="https://media.geeksforgeeks.org/wp-content/uploads/20230815003917/imgonline-com-ua-resize-OTGJBcAEfbktwNw-1024.jpg" alt="When | α | When | α | < 1, it is the case of Expansion
What is Time Shifting?
It is defined as the shifting of a signal along the time axis. In time shifting we relocate the signal left or right along the time axis. The general expression of time shifting is:
Time Shifting: x(t) →y(t) = x(t + k), k ≠0 where k is time in seconds
Depending on the value of k, there are two different cases for time shifting.
Case 1: When k > 0, means k is +ve. It is the case of time advance or Left shifting of the signal along the time axis.
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0, means k is +ve. It is the case of Left shifting” width=”1024″ height=”inherit” srcset=”https://media.geeksforgeeks.org/wp-content/uploads/20230819115418/3-(1).jpg 1100w,https://media.geeksforgeeks.org/wp-content/uploads/20230819115418/3-(1)-100.jpg 100w,https://media.geeksforgeeks.org/wp-content/uploads/20230819115418/3-(1)-200.jpg 200w,https://media.geeksforgeeks.org/wp-content/uploads/20230819115418/3-(1)-300.jpg 300w,https://media.geeksforgeeks.org/wp-content/uploads/20230819115418/3-(1)-660.jpg 660w,https://media.geeksforgeeks.org/wp-content/uploads/20230819115418/3-(1)-768.jpg 768w,https://media.geeksforgeeks.org/wp-content/uploads/20230819115418/3-(1)-1024.jpg 1024w”>When k > 0, means k is +ve. It is the case of Left shifting
Case 2: When k < 0, means k is -ve. It is the case of time delay or Right shifting of the signal along the time axis.
<img src="https://media.geeksforgeeks.org/wp-content/uploads/20230819115522/4-(1)-1024.jpg" alt="When k When k < 0, means k is -ve. It is the case of Right shifting
In time shifting, the amplitude of the signal remains the same, only shifting of signal occurs according to the value of k. The left or right shifting does not change the shape of the original signal.
What is Time Reversal?
It is defined as the special case of time scaling when the value of α is -1. In time reversal the new signal is the mirror image of the original signal about the y-axis. So it is also known as a Floating or reflection operation. The general expression of time reversal is:
Time Reversal: x(t) → y(t) = x(-t), where the value of α is -1
In time reversal, the amplitude of the signal remains the same, only the signal is changed to the mirror image of the original signal about the y-axis.
When α = – 1, it is the case of folding a signal along the y-axis.
Difference Between Time Scaling, Shifting, and Reversal
| It is defined as the compression or expansion of signal in time. |
It is defined as the shifting of the signal along the time axis. |
It is defined as the folding signal along the y-axis. |
|
Expression :
x(t)→y(t) = x(αt),
α ≠0
|
Expression :
x(t)→y(t) = x(t+k),
k ≠0
|
Expression :
x(t)→y(t) = x(-t),
α = -1
|
| Effect on Amplitude : There is no change in the amplitude of the signal |
Effect on Amplitude : There is no change in the amplitude of the signal |
Effect on Amplitude : There is no change in the amplitude of the signal |
| Effect on the shape of the original signal : Yes, due to compression and expansion of the signal, its shape gets affected. |
Effect on the shape of the original signal : No, due to left or right shifting, there is no effect on the shape of the original signal |
Effect on the shape of the original signal : The signal only gets reversed, so there is no effect on the shape of the original signal |
|
There are two cases of scaling:
Case1: When | α | > 1, it is the case of Compression.
Case 2: When | α | < 1, it is the case of Expansion.
|
There are two cases of scaling:
Case 1: When k > 0, means k is +ve. It is the case of left shifting.
Case 2: When k < 0, means k is +-ve. It is the case of right shifting.
|
There are no cases in time reversal because it is itself a special case of time scaling when α = -1. |
| It changes the frequent component due to variations in time. |
It does not directly affect the frequency of the signal. |
It is responsible for the change in phase of the signal. |
| It does not change the symmetry of the signal. |
It does not change the symmetry of the signal. |
It changes the symmetry of the signal, as reversing the signal about the y-axis can lead to a change in odd to even symmetry or vice-versa. |
| 2nd Priority order for sketching the signal |
1st Priority order for sketching the signal |
3rd Priority order for sketching the signal |
| It is applied to both CTS and DTS. |
It is applied to both CTS and DTS. |
It is applied to both CTS and DTS. |
Conclusion
At last, we conclude that all three terms, time scaling, time shifting, and time reversal are the basic operations that are useful to draw and analyze the signal after single or multiple operations. In this article, we have also learned some basic differences between them, and their effects on different parameters of the signal and discuss some cases as well. It is a very helpful tool to learn to get the correct output signal after sketching. Like time shifting, scaling, and reversal, there is a similar process for amplitude as well, in which we shift, scale, and reverse the amplitude of the given signal.
FAQs on Difference Between Time Shifting, Time Scaling and Time Reversal
Q:1. What is the priority order of operation during sketching a signal?
Answer:
The priority order of the signal is as follows:
Time Shifting → Time Scaling →Time Reversal
Note: It is not a factual order, but it can help you to solve sketching problem faster.
Q:2. Is time scaling, shifting, and reversal also valid for discrete-time signals?
Answer:
Yes, absolutely it is valid for both continuous-time signals as well as for discrete-time signals.
Q:3. What is the difference between Time scaling and Amplitude scaling?
Answer:
The main difference between Time sacling and amplitude scaling is that, in time scaling there is no change in amplitude of the signal whereas in amplitude scaling, the amplitude of the signal got changed.
Time Scaling: x(t) → y(t) = x(αt), α ≠0 where α is a constant.
Amplitude Scaling: x(t) → y(t) = βx(t), β ≠0 where β is a constant.
Q.4: Are time reversal and amplitude reversal is same because both show a folding effect?
Answer:
No, time reversal and amplitude reversal are not same, even though both show floading effect. Because in time reversal, new signal is fload or mirror image about y-axis whereas in amplitude reversal, new signal is fload or mirror about x-axis. And, as time reversal is a special case of time scaling , similarly amplitude reversal is a special case of amplitude scaling.
Time reversal: x(t)→y(t) = x(-t), α = -1 and Amplitude reversal: x(t)→y(t) = -x(t), β= -1
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