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Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE) PDF Download

Fourier Transform of Periodic signals.
We know the Fourier transform of the signal that assumes the value 1 identically is the dirac-delta function.

Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)
By the property of translation in the frequency domain, we get:

Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)

This is the result we will make use of in this section.
Suppose x(t) is a periodic signal with the period T, which admits a Fourier Series representation. Then,

Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Now since the Fourier transformation is linear, the above result can be used to obtain the Fourier Transform of the periodic signal x(t): 


Therefore,  Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)


By putting this transform in inverse Fourier transform equation, one can indeed confirm that one obtains back the Fourier series representation of x(t). 

Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)


Thus, the Fourier transform of a periodic signal having the Fourier series coefficients  is a train of impulses, occurring at multiples of the fundamental frequency,

the strength of the impulse at  being .


This looks like: 


  Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)

Basic Properties of Fourier Transform

Consider a signal x(t) with Fourier transform X(f). We'll see what happens to the Fourier transform of x(t) on time-reversal and conjugation. i.e: 

Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)

Now, we are aware that     Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)

 

Transform X'(f) of x(-t) is: .    Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)


Substitute Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)


Therefore, .Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)


Therefore, Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)


Applying this result to periodic signals (we have just seen their Fourier transform), you see that if  is the  Fourier Series co-efficient of a periodic signal x(t),  c -k is the K-th  Fourier series co-efficient of x(-t).


Now lets see how the Fourier Transform of  is related to that of x(t).

Starting with  Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)
taking conjugates, we get : Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)


Thus, Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)


And, therefore,Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)

 Applying this in the context of periodic signals, we see that if Cis the Kth Fourier Series co-efficient of a periodic signal x(t), then Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE) isthe kth Fourier series co-efficient of  Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)

Let us look at some simple consequences of these properties:

a) What can we say about the Fourier transform of an even signal x(t) (with Fourier transform X(f) ) ? x(-t) has Fourier transform X(-f). As x(t) is real, x(t) = x(-t), implying, X(f) = X(-f).
Thus, the Fourier transform of an even signal is even. Similarly, you can show the Fourier transform of an odd signal is odd.


b) What can we say about the Fourier transform of a real signal x(t), with Fourier transform X(f) ?
If x(t) is real,

Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE)

 

Conclusion: 

In this lecture you have learnt:

  • Fourier transformation is linear .
  • Fourier transform of x(-t) is X(-f).
  • Fourier transform of conjugate of x(t) is conjugate of X(-f).
  • The Fourier transform of an even signal is even
  • The Fourier transform of a real signal is Conjugate Symmetric .
The document Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Signals and Systems.
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FAQs on Fourier Transform of Periodic Signals & Some Basic Properties of Fourier Transform - Signals and Systems - Electrical Engineering (EE)

1. What is the Fourier Transform of a periodic signal?
Ans. The Fourier Transform of a periodic signal is a mathematical tool that decomposes the signal into its constituent frequencies. It represents the signal in the frequency domain, showing the amplitude and phase of each frequency component.
2. How is the Fourier Transform different from the Fourier Series?
Ans. The Fourier Transform is used for non-periodic signals, while the Fourier Series is specifically designed for periodic signals. The Fourier Series decomposes a periodic signal into a sum of sinusoids, while the Fourier Transform extends this concept to non-periodic signals by considering them as aperiodic and continuous over all time.
3. What are some basic properties of the Fourier Transform?
Ans. Some basic properties of the Fourier Transform include linearity, time shifting, frequency shifting, time scaling, and frequency scaling. Linearity states that the transform of a sum of signals is the sum of their individual transforms. Time shifting and frequency shifting involve translating the signal in the time or frequency domain, respectively. Time scaling and frequency scaling allow for compression or expansion of the signal in the time or frequency domain.
4. How is the Fourier Transform computed mathematically?
Ans. The Fourier Transform is computed using an integral formula. For a continuous-time signal, the transform can be found by integrating the product of the signal and a complex exponential function. The result is a complex-valued function of frequency. For a discrete-time signal, the transform can be found using a discrete version of the integral formula, which involves a summation instead of an integral.
5. What are some applications of the Fourier Transform?
Ans. The Fourier Transform has numerous applications in various fields. It is extensively used in signal processing, image processing, audio processing, communication systems, and many other areas. Some specific applications include noise filtering, image compression, audio equalization, spectral analysis, and channel estimation in wireless communication.
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