Derive the Fourier transform for the impulse train. The Fourier transform pair is given by (2.68) and (2.69).
The Correct Answer and Explanation is :
Fourier Transform of an Impulse Train
The impulse train, also known as the Dirac comb function, is defined as:
[
x_T(t) = \sum_{k=-\infty}^{\infty} \delta(t – kT)
]
where ( T ) is the period of the impulse train.
Step 1: Fourier Transform Definition
The Fourier transform of ( x_T(t) ) is given by:
[
X_T(f) = \int_{-\infty}^{\infty} x_T(t) e^{-j 2\pi f t} dt.
]
Step 2: Substituting the Impulse Train
Since ( x_T(t) ) is a sum of shifted Dirac delta functions, we substitute:
[
X_T(f) = \int_{-\infty}^{\infty} \sum_{k=-\infty}^{\infty} \delta(t – kT) e^{-j 2\pi f t} dt.
]
Using the sifting property of the delta function:
[
\int_{-\infty}^{\infty} \delta(t – kT) e^{-j 2\pi f t} dt = e^{-j 2\pi f kT}.
]
Thus, the Fourier transform simplifies to:
[
X_T(f) = \sum_{k=-\infty}^{\infty} e^{-j 2\pi f kT}.
]
Step 3: Recognizing the Result
This summation represents the Fourier series of another impulse train in the frequency domain. Using the Poisson summation formula, we obtain:
[
X_T(f) = \frac{1}{T} \sum_{n=-\infty}^{\infty} \delta \left( f – \frac{n}{T} \right).
]
Thus, the Fourier transform of an impulse train in the time domain is another impulse train in the frequency domain, but scaled by ( 1/T ) and spaced at integer multiples of ( 1/T ):
[
\sum_{k=-\infty}^{\infty} \delta(t – kT) \quad \xleftrightarrow{\mathcal{F}} \quad \frac{1}{T} \sum_{n=-\infty}^{\infty} \delta \left( f – \frac{n}{T} \right).
]
Explanation (300 Words)
The impulse train, ( x_T(t) ), consists of delta functions spaced by ( T ) in the time domain. Its Fourier transform can be derived by directly applying the definition.
Since the Dirac delta function has a key property that integrates to 1 when sampled at its argument, we exploit this property while computing the Fourier transform. When we apply the Fourier transform integral to the sum of delta functions, each delta function contributes an exponential term evaluated at its shift.
The result is an infinite sum of exponentials, which corresponds to another impulse train in the frequency domain. However, this frequency-domain impulse train is scaled by ( 1/T ) and has impulses located at integer multiples of ( 1/T ).
This result is fundamental in signal processing. It indicates that sampling a continuous signal at intervals of ( T ) produces a repetition of its spectrum in the frequency domain. This principle is the basis of the sampling theorem and explains the occurrence of aliasing when sampling is not properly done.
Thus, the Fourier transform of an impulse train is itself an impulse train, emphasizing the duality between periodicity in one domain and discreteness in the other.
Now, let me generate an image representing this Fourier transform relationship.
Here is the visualization of the Fourier transform of an impulse train, showing the time-domain and frequency-domain representations. Let me know if you need further clarifications!
