Fourier Transform of Periodic signals.
We know the Fourier transform of the signal that assumes the value 1 identically is the dirac-delta function.
By the property of translation in the frequency domain, we get:
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,
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,
By putting this transform in inverse Fourier transform equation, one can indeed confirm that one obtains back the Fourier series representation of x(t).
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:
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:
Now, we are aware that
Transform X'(f) of x(-t) is: .
Substitute
Therefore, .
Therefore,
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
taking conjugates, we get :
Thus,
And, therefore,
Applying this in the context of periodic signals, we see that if Ck is the Kth Fourier Series co-efficient of a periodic signal x(t), then isthe kth Fourier series co-efficient of
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,
Conclusion:
In this lecture you have learnt:
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1. What is the Fourier Transform of a periodic signal? |
2. How is the Fourier Transform different from the Fourier Series? |
3. What are some basic properties of the Fourier Transform? |
4. How is the Fourier Transform computed mathematically? |
5. What are some applications of the Fourier Transform? |
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