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Course by: Thad Welch. E-mail the author

DFT Basis

Module by: Richard Baraniuk. E-mail the author

• Summary of the basis points.
• An ONB for CN N signals constructed from:
• complex sinusoids
• that are all N-periodic when considered as ∞-length signals.

Note:

There are only N unique sinusoids like this in CN N .
General formula:
n ,0nN1: b k n=1Nei(2πNk)n n 0 n N 1 b k n 1 N 2 N k n
(1)

Example 1

k=0 k 0 : DC component

k=0 k 0 : 1 cycle k=2 k 2 : 2 cycles

Example 2

What is the highest frequency of oscillation (the most wiggles)?

b N / 2 b N / 2 : N2 N 2 cycles

What happens for N2<k<N1 N 2 k N 1 ?

The number of wiggles starts decreasing!

Note:

b k b r b k b r , rk r k , k ,: b k 2=1 k 2 b k 1

In other words, b k b k forms an ONB for CN N x=Bα x B α α=BHx α B x xn=1N k =0N1αkei2πNkn x n 1 N k 0 N 1 α k 2 N k n αn=1N k =0N1xke(i2πNkn) α n 1 N k 0 N 1 x k 2 N k n

Note:

Most DSP texts (as well as matlab) work with unormalized DFT basis vectors: b ' k n=ei2πNkn b ' k n 2 N k n . Notice that there is not a 1N 1 N factor.
The formulas change as follows: xn=1N k =0N1 α ' kei2πNkn x n 1 N k 0 N 1 α ' k 2 N k n α ' n= k =0N1xke(i2πNkn) α ' n k 0 N 1 x k 2 N k n

Exercise 1

How do the Parseval and Plancharel formulaii change?

DFT Properties

x=BX x B X X=BHx X B x

Table 1: DFT Properties
Finite-length Sequences (Length N) N-Point DFT (Length N)
1. xn x n Xk X k
2. x 1 n x 1 n , x 2 n x 2 n X 1 k X 1 k , X 2 k X 2 k
3. a x 1 n+b x 2 n a x 1 n b x 2 n a X 1 k+b X 2 k a X 1 k b X 2 k
4. Xn X n x(k)modN x k N
5. x(nm)modN x n m N Xkei2πNkn X k 2 N k n
6. WmodN(ln)xn W N l n x n X(kl)modN X k l N
7. m =0N1 x 1 m x 2 (nm)modN m 0 N 1 x 1 m x 2 n m N X 1 k X 2 k X 1 k X 2 k
8. x 1 n x 2 n x 1 n x 2 n m =0N1 X 1 l X 2 (kl)modN m 0 N 1 X 1 l X 2 k l N
9. xn¯ x n X(k)modN¯ X k N
10. x(n)modN¯ x n N Xk¯ X k
11. xn x n X ep k=12(XkmodN+X(k)modN¯) X ep k 1 2 X k N X k N
12. ixn x n X op k=12(XkmodNX(k)modN¯) X op k 1 2 X k N X k N
13. x ep n=12(xnmodN+x(n)modN¯) x ep n 1 2 x n N x n N Xk X k
14. x op n=12(XnmodNx(n)modN¯) x op n 1 2 X n N x n N iXk X k
15. Symmetry Properties
 X⁢k=X⁢(−k)modN¯ X k X k N ℜ⁢X⁢k=ℜ⁢X⁢(−k)modN X k X k N ℑ⁢X⁢k=−ℑ⁢X⁢(−k)modN X k X k N |X⁢k|=|X⁢(−k)modN| X k X k N ∡⁢X⁢k=−∡⁢X⁢(−k)modN ∡ X k ∡ X k N
16. x ep n=12(xn+x(n)modN) x ep n 1 2 x n x n N Xk X k
17. x op n=12(xnx(n)modN) x op n 1 2 x n x n N Xk X k

Note:

Properties 15-17 apply only when xn x n is real.

Complex Sinusoids in a nutshell

A complex sinusoid is denoted as n,k ,0n,kN1:ei2πNkn n,k 0 n,k N 1 2 N k n where nn is the time index and kk is the frequency index. The same equation with frequency, ω k ,0 ω k 2π: ω k =2πNk ω k 0 ω k 2 ω k 2 N k , is written as: ei ω k n ω k n C sinusoid proceeds CCW around the unit circle (because |ei ω k n|=1 ω k n 1 ) with step size (in angle) ω k ω k .

Example 3

C sinusoid makes exactly kk revolutions of the circle and repeats when n=N n N at the n=0 n 0 spot.

Read off real and imaginary parts: ei ω k n=cos ω k n+isin ω k n ω k n ω k n ω k n by motion along real and imaginary axes.

Consider the following examples:

Example 5

Notice that for n=3 n 3 , the plot is exactly like the one for k=1 k 1 in Example 4, but moves backwards (CW) around the circle.

Negative frequency is nothing bizarre; it just means that we move around the circle CW rather than CCW. Figure 15 shows how to do this for N=8 N 8 .

In general, this is easy to show: ei2πNkn=ei2πNknei2πNN=ei2πN(k+N)n 2 N k n 2 N k n 2 N N 2 N k N n For example, for N=8 N 8 , k=-1 k -1 and k=-1+8=7 k -1 8 7 yield the same sinusoid.

Summary: Frequency

ei2πNkn=ei ω k n 2 N k n ω k n
(2)
ω k =2πNk ω k 2 N k

Impact on DFT

Matlab's fftshift command swaps left and right halfs of a vector:

DFT Notation

x=Bα x B α α=BHx α B x Now replace αα by capital XX: x=BX x B X X=BHx X B x xn DFT Xk x n DFT X k

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