DFT of x[n] is defined as

The Discrete Fourier Transform (DFT) of a sequence x[n] is defined as follows:

1. Given a sequence x[n] of length N, where n ranges from 0 to N-1, the DFT of x[n] is computed using the formula:

X[k] = ∑(n=0 to N-1) x[n] * e^(-j2πkn/N)

where X[k] represents the kth frequency component of the DFT and j is the imaginary unit.

2. To calculate X[k], we iterate over all values of k from 0 to N-1. For each k, we compute the sum of x[n] multiplied by the complex exponential term e^(-j2πkn/N) for all values of n.

3. The resulting X[k] values represent the frequency components of the input sequence x[n]. The magnitude of X[k] gives the amplitude of the corresponding frequency component, while the phase of X[k] represents the phase shift of that component.

4. The DFT can be represented as a complex-valued sequence X[k], where each element X[k] is a complex number with a real and imaginary part. Alternatively, we can represent the DFT as a magnitude spectrum (|X[k]|) and a phase spectrum (arg(X[k])).

5. The DFT is a powerful tool in signal processing and is commonly used for tasks such as spectral analysis, filtering, and signal synthesis. It allows us to analyze the frequency content of a discrete-time signal and extract useful information from it.

Note: The DFT assumes that the input sequence x[n] is periodic with a period of N. If the input sequence is not periodic, it can be zero-padded to a length of N before computing the DFT to avoid spectral leakage.