Question

Determine the DFT of the following sequences:

a.) x[n]=cos(2π⅛n) for n=[0,1,…,7]
b.) y₁=[0,−1,0,1]
c.) y₂=[j,0,j,1]

Answer 1

The student wishes to determine the Discrete Fourier Transform (DFT) of three sequences: a cosine wave and two shorter sample sequences including one with complex numbers.

Step-by-step explanation:

The student is asking to determine the Discrete Fourier Transform (DFT) of different sequences. A DFT is a mathematical transformation used in signal processing, which converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform which is essentially a frequency domain representation.

For the sequence x[n]=cos(2π⅛ n) for n=[0,1,…,7]:

This is a real-valued function representing a cosine wave with a period that fits within the 8 samples. The DFT will have two non-zero coefficients corresponding to the frequencies at which the cosine function oscillates.

For the sequences y1=[0,−1,0,1] and y2=[j,0,j,1]:

These are shorter sequences with 4 samples each. y1 is a simple waveform with 0 amplitude at the ends and alternating values in between. y2 contains complex numbers, where 'j' represents the imaginary unit. The DFT for these sequences will show the frequencies and phase information for the waves represented by these samples.