Show Parseval’s theorem for the DFT

Show Parseval’s theorem for the DFT given in Table 7.4 and use it to prove Parseval’s relation given in the same Table.

2. The 2D-DFT introduced in Problem 26 possesses properties that are similar to those of 1D-DFT.

The Correct Answer and Explanation is :

Parseval’s theorem for the Discrete Fourier Transform (DFT) states that the total energy of a discrete-time signal is equal to the total energy of its DFT. Mathematically, this is expressed as:

[ \sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X[k]|^2 ]

where ( x[n] ) is the original signal, ( X[k] ) is its DFT, and ( N ) is the number of samples.

Proof:

1. Definition of DFT: The DFT of a sequence ( x[n] ) is given by: [ X[k] = \sum_{n=0}^{N-1} x[n] e^{-i \frac{2\pi}{N} k n} ]
2. Magnitude Squared of ( X[k] ): The magnitude squared of ( X[k] ) is: [ |X[k]|^2 = X[k] X^[k] = \left( \sum_{n=0}^{N-1} x[n] e^{-i \frac{2\pi}{N} k n} \right) \left( \sum_{m=0}^{N-1} x[m]^ e^{i \frac{2\pi}{N} k m} \right) ]
3. Expanding the Product: Expanding the product of sums: [ |X[k]|^2 = \sum_{n=0}^{N-1} \sum_{m=0}^{N-1} x[n] x[m]^* e^{-i \frac{2\pi}{N} k (n – m)} ]
4. Summing Over ( k ): Summing both sides over ( k ) from 0 to ( N-1 ): [ \sum_{k=0}^{N-1} |X[k]|^2 = \sum_{k=0}^{N-1} \sum_{n=0}^{N-1} \sum_{m=0}^{N-1} x[n] x[m]^* e^{-i \frac{2\pi}{N} k (n – m)} ]
5. Evaluating the Inner Sum: The inner sum is a geometric series: [ \sum_{k=0}^{N-1} e^{-i \frac{2\pi}{N} k (n – m)} = N \delta_{n,m} ] where ( \delta_{n,m} ) is the Kronecker delta function, which is 1 when ( n = m ) and 0 otherwise.
6. Simplifying the Expression: Using the Kronecker delta: [ \sum_{k=0}^{N-1} |X[k]|^2 = \sum_{n=0}^{N-1} \sum_{m=0}^{N-1} x[n] x[m]^* N \delta_{n,m} ] [ = N \sum_{n=0}^{N-1} |x[n]|^2 ]
7. Final Step: Dividing both sides by ( N ): [ \sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X[k]|^2 ]

This completes the proof of Parseval’s theorem for the DFT.

Explanation:

Parseval’s theorem establishes a fundamental relationship between a signal and its frequency components. It asserts that the total energy (or power) of a discrete-time signal is conserved when transformed into the frequency domain via the DFT. This conservation is crucial in signal processing, as it ensures that no information is lost during the transformation between time and frequency representations.

In practical terms, Parseval’s theorem allows engineers and scientists to analyze the energy distribution of signals in the frequency domain, which is essential for applications like audio processing, telecommunications, and image analysis. By understanding how energy is distributed across different frequency components, one can design filters, compress signals, and perform other operations that manipulate the frequency content of signals without altering their total energy.

Moreover, Parseval’s theorem is a special case of the more general Plancherel theorem, which applies to Fourier transforms in various contexts. The theorem’s proof relies on the orthogonality of the complex exponential functions used in the DFT, which form an orthonormal basis for the space of square-integrable sequences. This orthogonality property ensures that the inner product (and thus the energy) is preserved under the DFT transformation.

Understanding and applying Parseval’s theorem is fundamental for anyone working with Fourier analysis and its applications, as it provides a theoretical foundation for energy conservation in signal processing.

For a visual explanation and further insights into Parseval’s theorem, you might find the following video helpful:

videoParseval’s Theorem – YouTubeturn0search12