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Transformation on Independent Variables | Signals & Systems | GATE Preparation | EC
Real Life Application
This concept finds application in data transmission and reception in TV signals, internet and various signal processing applications. Ultra slow motion is one application area of time scaling operation.
Transformation on Independent Variable (time) is a very important concept in Signals and Systems. It finds its application in Convolution, Periodic signals, Symmetric signals, to verify systems as time invariant systems and also in various transformation techniques like Fourier series, Fourier Transform, Laplace and Z transform.
There are three operations that are performed as part of transformation on time. They are
- Time Shifting
- Time Scaling
- Time Reversal
Time shifting has two subparts: Time-Advance and Time Delay. For a signal f(t), f(t+t’) is time advanced signal and f(t-t’) is time delayed signal where t’ >0. Relation between Signals at transmitter and receiver have a relation of time shift provided the signal is noise free.
Time Scaling also has two Subparts: Time-Compression and Time-Expansion. For a signal f(t), f(at) is time scaled version of it where ‘a’ is scaling parameter, a>0. f(at) represents the time-compressed version when a>1 and time expanded version when a By means of time-scaling, information is not distorted.
Time- Reversal of a signal represented by f(-t) is the reflected version of original signal f(t). After learning all the three operations one by one, now its time to apply all the three operations simultaneously on the signal. For example, On the signal f (t), we have to apply transformations to form f(-at±b). The best way of doing it is to apply Shift then Scale and reversal at last. The reason why it is best is because the chance of doing a mistake to get final transformed signal is less. Having said that, its also important to learn about other permutations of doing it. i.e. Scale, reversal and Shift or reversal, shift and scale etc. But the care to be taken is whenever shift is done after reversal or scale or both. This has been explained clearly in the class.
Whatever transformation has been done to time, the same can be applied to frequency as well as in frequency domain the independent variable is frequency.
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