- State and prove Parseval’s theorem for complex exponential Fourier series.
2.Distinguish between the exponential form of Fourier series and Fourier transform. What is the
nature of the “transform pairs” in the above two cases? How would you obtain Fourier integral from Fourier series? Define Fourier transform pair. Define the Fourier transform of a time function and explain under what conditions it exists.
The Correct Answer and Explanation is :
1. Parseval’s Theorem for Complex Exponential Fourier Series:
Parseval’s Theorem for a complex exponential Fourier series states that the total energy of a periodic signal, as measured in the time domain, is equal to the total energy in the frequency domain.
If ( x(t) ) is a periodic signal with period ( T ), and its Fourier series expansion is given by:
[
x(t) = \sum_{n=-\infty}^{\infty} c_n e^{j n \omega_0 t}
]
where ( \omega_0 = \frac{2 \pi}{T} ) is the fundamental frequency and ( c_n ) are the complex Fourier coefficients, then Parseval’s theorem states:
[
\frac{1}{T} \int_0^T |x(t)|^2 \, dt = \sum_{n=-\infty}^{\infty} |c_n|^2
]
This means that the total power (or energy) of the signal in the time domain is equal to the sum of the squared magnitudes of the Fourier coefficients in the frequency domain.
Proof:
- Starting with the Fourier series representation of ( x(t) ):
[
x(t) = \sum_{n=-\infty}^{\infty} c_n e^{j n \omega_0 t}
]
- To compute the energy of the signal, we calculate the integral of the squared modulus:
[
\frac{1}{T} \int_0^T |x(t)|^2 dt
]
- Expanding ( |x(t)|^2 ), we get:
[
|x(t)|^2 = \left| \sum_{n=-\infty}^{\infty} c_n e^{j n \omega_0 t} \right|^2
]
- Using the orthogonality property of the complex exponentials ( e^{j n \omega_0 t} ), we can show that the cross terms vanish, leaving:
[
\frac{1}{T} \int_0^T |x(t)|^2 dt = \sum_{n=-\infty}^{\infty} |c_n|^2
]
Thus, the energy in the time domain is equal to the sum of the squared magnitudes of the Fourier coefficients.
2. Distinction Between Exponential Form of Fourier Series and Fourier Transform:
- Exponential Form of Fourier Series: This represents periodic functions. The Fourier series decomposes a periodic signal ( x(t) ) into a sum of complex exponentials of discrete frequencies ( n \omega_0 ) where ( \omega_0 = \frac{2\pi}{T} ) is the fundamental frequency. The coefficients ( c_n ) represent the contribution of each frequency component.
- Fourier Transform: The Fourier transform, on the other hand, is used to represent non-periodic signals as an integral of complex exponentials over a continuous frequency range. It gives a spectrum of all possible frequencies that make up the signal. It is defined as: [
X(f) = \int_{-\infty}^{\infty} x(t) e^{-j 2 \pi f t} \, dt
]
Nature of Transform Pairs:
- In the case of the Fourier series, the transform pair consists of the signal ( x(t) ) and its Fourier coefficients ( c_n ), where the coefficients are discrete.
- In the Fourier transform case, the transform pair consists of the signal ( x(t) ) and its transform ( X(f) ), where the frequency variable ( f ) is continuous.
Derivation of Fourier Integral from Fourier Series:
To obtain the Fourier integral from the Fourier series, consider a periodic function with period ( T ). Let ( x(t) ) be approximated by a periodic sum of sines and cosines. As the period ( T ) becomes infinite, the discrete frequencies in the Fourier series become a continuous range of frequencies, leading to the Fourier transform. Mathematically, this is shown as:
[
\lim_{T \to \infty} \sum_{n=-\infty}^{\infty} c_n e^{j n \omega_0 t} \to \int_{-\infty}^{\infty} X(f) e^{j 2 \pi f t} \, df
]
Fourier Transform Pair:
The Fourier transform pair consists of a time-domain function ( x(t) ) and its frequency-domain counterpart ( X(f) ), such that:
- ( X(f) = \int_{-\infty}^{\infty} x(t) e^{-j 2 \pi f t} \, dt )
- ( x(t) = \int_{-\infty}^{\infty} X(f) e^{j 2 \pi f t} \, df )
Fourier Transform of a Time Function:
The Fourier transform of a time-domain function ( x(t) ) is given by:
[
X(f) = \int_{-\infty}^{\infty} x(t) e^{-j 2 \pi f t} \, dt
]
It exists under certain conditions, most commonly when ( x(t) ) is absolutely integrable, i.e., ( \int_{-\infty}^{\infty} |x(t)| \, dt < \infty ). For practical signals, the Fourier transform may also exist in the sense of distributions if ( x(t) ) is not strictly integrable but satisfies certain regularity conditions.