Operation on signals - Independent variables
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This document discusses basic operations that can be performed on signals based on their independent variables, including time shifting, time scaling, and time reversal. Time shifting involves delaying or advancing a signal along the time axis without changing other characteristics. Time scaling compresses (decimation) or expands (interpolation) a signal by multiplying the time variable. Time reversal reflects a signal across the vertical axis by multiplying the time variable by negative one. These operations have applications in areas like signal processing, speech processing, image processing, and non-destructive testing.
![TIME SHIFTING - DELAY
Here the original signal, x[n], spans from n = -3 to n = 3 and has the values -2, 0, 1, -
3, 2, -1, and 3, as shown in Figure
Suppose that we want to move this signal right by three units (i.e., we want a new
signal whose amplitudes are the same but are shifted right three times).
Here the original signal, x[n], spans from n = -3 to n = 3 and has the values -2, 0, 1, -
3, 2, -1, and
This means that we desire our output signal y[n] to span from n = 0to n = 6. Such a
signal is shown as Figure and can be mathematically written as y[n] = x[n-3].
This kind of signal is referred to as time-delayed because we have made the signal
arrive three units late.](https://image.slidesharecdn.com/unit-i-6-200817061042/85/Operation-on-signals-Independent-variables-4-320.jpg)
![TIME SHIFTING - ADVANCE
On the other hand, let's say that we want the same signal to arrive early.
Consider a case where we want our output signal to be advanced by, say, two
units.
This objective can be accomplished by shifting the signal to the left by two time
units, i.e., y[n] = x[n+2].
The corresponding input and output signals are shown in Figure.
Our output signal has the same values as the original signal but spans from n = -
5 to n = 1 instead of n = -3 to n = 3.
The signal shown in Figure is aptly referred to as a time-advanced signal.](https://image.slidesharecdn.com/unit-i-6-200817061042/85/Operation-on-signals-Independent-variables-6-320.jpg)
![TIME SCALING - DECIMATION
In decimation, the input of the signal is changed to be f[cn] .
The quantity used for decimation cc must be an integer so that the input
takes values for which a discrete function is properly defined.
The decimated signal f[cn] corresponds to the original signal f[n] where
only each n sample is preserved (including f[0]), and so we are throwing
away samples of the signal (or decimating it).](https://image.slidesharecdn.com/unit-i-6-200817061042/85/Operation-on-signals-Independent-variables-10-320.jpg)
![TIME SCALING - EXPANSION
In expansion, the input of the signal is changed to be f[n/c].
We know that the signal f[n] is defined only for integer values of the
input n.
Thus, in the expanded signal we can only place the entries of the
original signal f at values of n that are multiples of c.
In other words, we are spacing the values of the discrete-time
signal c−1 entries away from each other.
Since the signal is undefined elsewhere, the standard convention in
expansion is to fill in the undetermined values with zeros.](https://image.slidesharecdn.com/unit-i-6-200817061042/85/Operation-on-signals-Independent-variables-12-320.jpg)
![TIME REVERSAL
Until now, we have assumed our independent variable representing the
signal to be positive. Why should this be the case? Can't it be negative?
It can be negative. In fact, one can make it negative just by multiplying
it by -1. This causes the original signal to flip along its y-axis. That is, it
results in the reflection of the signal along its vertical axis of reference.
As a result, the operation is aptly known as the time reversal or time
reflection of the signal
For example, let's consider our input signal to be x[n], shown in Figure.
The effect of substituting –n in the place of n results in the signal y[n] as
shown in Figure.](https://image.slidesharecdn.com/unit-i-6-200817061042/85/Operation-on-signals-Independent-variables-15-320.jpg)
Operation on signals - Independent variables
- 1. 15EE55C – DIGITAL SIGNAL PROCESSING AND ITS APPLICATIONS BASIC OPERATIONS ON SIGNALS – INDEPENDENT VARIABLES Dr. M. Bakrutheen AP(SG)/EEE Mr. K. Karthik Kumar AP/EEE DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING NATIONAL ENGINEERING COLLEGE, K.R. NAGAR, KOVILPATTI – 628 503 (An Autonomous Institution, Affiliated to Anna University – Chennai)
- 2. BASIC OPERATIONS ON SIGNALS Basic signal operations categorized into two types depending on whether they operated on dependent or independent variable(s) representing the signals. Addition, subtraction, multiplication fall under the category of basic signal operations acting on the dependent variable. Time shifting, time scaling and time reversal, which manipulate the signal characteristics by acting on the independent variable(s) fall under the category basic operation depends on the independent variable.
- 3. TIME SHIFTING Suppose that we have a signal x(n) and we define a new signal by adding/subtracting a finite time value to/from it. We now have a new signal, y(n). The mathematical expression for this would be x(n ± N). Graphically, this kind of signal operation results in a positive or negative “shift” of the signal along its time axis. However, note that while doing so, none of its characteristics are altered. This means that the time-shifting operation results in the change of just the positioning of the signal without affecting its amplitude or span
- 4. TIME SHIFTING - DELAY Here the original signal, x[n], spans from n = -3 to n = 3 and has the values -2, 0, 1, - 3, 2, -1, and 3, as shown in Figure Suppose that we want to move this signal right by three units (i.e., we want a new signal whose amplitudes are the same but are shifted right three times). Here the original signal, x[n], spans from n = -3 to n = 3 and has the values -2, 0, 1, - 3, 2, -1, and This means that we desire our output signal y[n] to span from n = 0to n = 6. Such a signal is shown as Figure and can be mathematically written as y[n] = x[n-3]. This kind of signal is referred to as time-delayed because we have made the signal arrive three units late.
- 5. TIME SHIFTING - DELAY
- 6. TIME SHIFTING - ADVANCE On the other hand, let's say that we want the same signal to arrive early. Consider a case where we want our output signal to be advanced by, say, two units. This objective can be accomplished by shifting the signal to the left by two time units, i.e., y[n] = x[n+2]. The corresponding input and output signals are shown in Figure. Our output signal has the same values as the original signal but spans from n = - 5 to n = 1 instead of n = -3 to n = 3. The signal shown in Figure is aptly referred to as a time-advanced signal.
- 7. TIME SHIFTING - ADVANCE
- 8. TIME SHIFTING – PRACTICAL SCENARIO Time-shifting is an important operation that is used in many signal-processing applications. For example, a time-delayed version of the signal is used when performing autocorrelation. (You can learn more about autocorrelation in my previous article, understanding correlation). Another field that involves the concept of time delay is artificial intelligence, such as in systems that use time delay neural networks.
- 9. TIME SCALING Time scaling compresses or dilates a signal by multiplying the time variable by some quantity. If that quantity is greater than one, the signal becomes narrower and the operation is called decimation. In contrast, if the quantity is less than one, the signal becomes wider and the operation is called expansion or interpolation, depending on how the gaps between values are filled.
- 10. TIME SCALING - DECIMATION In decimation, the input of the signal is changed to be f[cn] . The quantity used for decimation cc must be an integer so that the input takes values for which a discrete function is properly defined. The decimated signal f[cn] corresponds to the original signal f[n] where only each n sample is preserved (including f[0]), and so we are throwing away samples of the signal (or decimating it).
- 11. TIME SCALING - DECIMATION
- 12. TIME SCALING - EXPANSION In expansion, the input of the signal is changed to be f[n/c]. We know that the signal f[n] is defined only for integer values of the input n. Thus, in the expanded signal we can only place the entries of the original signal f at values of n that are multiples of c. In other words, we are spacing the values of the discrete-time signal c−1 entries away from each other. Since the signal is undefined elsewhere, the standard convention in expansion is to fill in the undetermined values with zeros.
- 13. TIME SCALING - EXPANSION
- 14. TIME SCALING – PRACTICAL SCENARIO Basically, when we perform time scaling, we change the rate at which the signal is sampled. Changing the sampling rate of a signal is employed in the field of speech processing. A particular example of this would be a time-scaling-algorithm-based system developed to read text to the visually impaired Next, the technique of interpolation is used in Geodesic applications. This is because, in most of these applications, one will be required to find out or predict an unknown parameter from a limited amount of available data.
- 15. TIME REVERSAL Until now, we have assumed our independent variable representing the signal to be positive. Why should this be the case? Can't it be negative? It can be negative. In fact, one can make it negative just by multiplying it by -1. This causes the original signal to flip along its y-axis. That is, it results in the reflection of the signal along its vertical axis of reference. As a result, the operation is aptly known as the time reversal or time reflection of the signal For example, let's consider our input signal to be x[n], shown in Figure. The effect of substituting –n in the place of n results in the signal y[n] as shown in Figure.
- 16. TIME REVERSAL
- 17. TIME REVERSAL – PRACTICAL SCENARIO Time reversal is an important preliminary step when computing the convolution of signals: one signal is kept in its original state while the other is mirror-imaged and slid along the former signal to obtain the result. Time-reversal operations, therefore, are useful in various image-processing procedures, such as edge detection A time-reversal technique in the form of the time reverse numerical simulation (TRNS) method can be effectively used to determine defects. For example, the TRNS method aids in finding out the exact position of a notch which is a part of the structure along which a guided wave propagates