UNIT 7 | unit 7 discrete fourier series

DSP

UNIT 7

DFS

7.1 Discrete Fourier series

A real, N-periodic, discrete-time signal x[n] can be represented by a linear combination of the complex exponential signals

In these expressions, , and the discrete-time fundamental frequency is . This discrete-time Fourier series representation provides notions of frequency content of discrete-time signals, and it is very convenient for calculations involving linear, time-invariant systems because complex exponentials are eigenfunctions of LTI systems.

The complex coefficients can be calculated from the expression

The are called the spectral coefficients of the signal x[n]. A plot of k is called the magnitude spectrum of x[n], and a plot of k is called the phase spectrum of x[n]. These plots, particularly the magnitude spectrum, provide a picture of the frequency composition of the signal. Notice that the spectral coefficients repeat as k is varied. In particular, for any value of k,

7.2 Discrete Fourier transform

The discrete Fourier transform transforms a sequence of N complex numbers into another sequence of complex numbers

, which is defined by

Example:

Let N =4 and

Here we demonstrate how to calculate the DFT of x using equation (1)

7.3 Properties of DFT, FFT

Properties of DFT

Linear Property

If x (t) -> X (w)

Y (t) -> Y (w) then

a x (t) + b y (t) -> a X (w) +b Y (w)

Time Shifting Property

If x (t) ⟷F.TX (ω)

Then Time shifting property states that

x (t−t0)⟷F.T e−jω0t X(ω)

Frequency Shifting Property

If x (t) ⟷X (ω)

Then frequency shifting property states that

Ejω0t.x (t) ⟷X (ω−ω0)

Time Reversal Property

If x (t) ⟷X (ω)

Then Time reversal property states that

x (−t)⟷X(−ω)

Differentiation Property

If x (t) ⟷X (ω)

Then Differentiation property states that

Dx (t) dt ⟷jω.X(ω)

dnx (t)dtn⟷(jω)n.X(ω)

Integration Property

Integration property states that

∫x (t) dt⟷1jω X (ω)

Then

∭...∫x (t) dt ⟷ (jω) n X (ω)

Multiplication and Convolution Properties

If x (t) ⟷X (ω)

y (t)⟷Y (ω)

Then multiplication property states that

x (t).y (t)⟷X (ω) ∗Y (ω)

And convolution property states that

x (t)∗y(t)⟷1/2πX(ω).Y(ω)

Numericals:

Compute the N-point DFT of x (n) =3 δ (n).

2. Compute the N-point DFT of x(n)=7(n−n0)

Solution −We know that,

Substituting the value of x(n),

Properties of FFT

Certain algorithms permit implementations of Discrete Fourier transform with considerable savings in computation time. These algorithms are known as Fast Fourier Transform.

FFT algorithm are based on fundamental principle of decomposing the computation of DFT of sequence length into successively smaller discrete Fourier transforms.

They are basically two classes of FFT algorithms

Decimation in time
Decimation in frequency

Decimation-in-time algorithm

This algorithm is known as Radix-2 DIT-FFT algorithm which means the number of output points N can be expressed as a power of 2 that is N = 2M where M is an integer.

Let x(n) be a sequence where N is assumed to be a power of 2. Decimate or break this sequence into two sequences of length N/2 where one sequence consists of even-indexed values of x(n) and the other of odd-indexed values of x(n).

Xe(n) = x(2n) n= 0,1,……….. N/2-1

Xo(n) = x(2n+1) n= 0,1,……….. N/2 -1

The N-pt DFT of x(n) can be written as

X(k) = WN nk where k=0,1……….. N-1.