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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 ,0≤n≤N−1: b k ⁢n=1N⁢ei⁢(2⁢πN⁢k)⁢n n 0 n N 1 b k n 1 N 2 N k n
(1)

Example 1

k=0 k 0 : DC component

Figure 1: b 0 ⁢n=1N b 0 n 1 N
Figure 1 (DFT1.png)
k=0 k 0 : 1 cycle
Figure 2: ℜ⁢ b 1 ⁢n=cos2⁢πN⁢k⁢nN b 1 n 2 N k n N
Figure 2 (DFT2.png)
Figure 3: ℑ⁢ b 1 ⁢n=sin2⁢πN⁢k⁢nN b 1 n 2 N k n N
Figure 3 (DFT3.png)
k=2 k 2 : 2 cycles
Figure 4: ℜ⁢ b 2 b 2
Figure 4 (DFT4.png)
Figure 5: ℑ⁢ b 2 b 2
Figure 5 (DFT5.png)

Example 2

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

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

Figure 6: ℜ⁢ b N / 2 ⁢n=1N⁢cosπ⁢n b N / 2 n 1 N n
Figure 6 (DFT6.png)
Figure 7: ℑ⁢ b N / 2 ⁢n=1N⁢sinπ⁢n=0 b N / 2 n 1 N n 0
Figure 7 (DFT7.png)
What happens for N2<k<N−1 N 2 k N 1 ?

The number of wiggles starts decreasing!

Figure 8
Figure 8 (DFT8.png)

Note:

b k ⊥ b r ⊥ b k b r , r≠k 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=B⁢α x B α α=BH⁢x α B x x⁢n=1N⁢∑ k =0N−1α⁢k⁢ei⁢2⁢πN⁢k⁢n x n 1 N k 0 N 1 α k 2 N k n α⁢n=1N⁢∑ k =0N−1x⁢k⁢e−(i⁢2⁢πN⁢k⁢n) 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=ei⁢2⁢πN⁢k⁢n b ' k n 2 N k n . Notice that there is not a 1N 1 N factor.
The formulas change as follows: x⁢n=1N⁢∑ k =0N−1 α ' ⁢k⁢ei⁢2⁢πN⁢k⁢n x n 1 N k 0 N 1 α ' k 2 N k n α ' ⁢n=∑ k =0N−1x⁢k⁢e−(i⁢2⁢πN⁢k⁢n) 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=B⁢X x B X X=BH⁢x X B x

Table 1: DFT Properties
Finite-length Sequences (Length N) N-Point DFT (Length N)
1. x⁢n x n X⁢k 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. X⁢n X n x⁢(−k)modN x k N
5. x⁢(n−m)modN x n m N X⁢k⁢ei⁢2⁢πN⁢k⁢n X k 2 N k n
6. WmodN−(l⁢n)⁢x⁢n W N l n x n X⁢(k−l)modN X k l N
7. ∑ m =0N−1 x 1 ⁢m⁢ x 2 ⁢(n−m)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 =0N−1 X 1 ⁢l⁢ X 2 ⁢(k−l)modN m 0 N 1 X 1 l X 2 k l N
9. x⁢n¯ x n X⁢(−k)modN¯ X k N
10. x⁢(−n)modN¯ x n N X⁢k¯ X k
11. ℜ⁢x⁢n x n X ep ⁢k=12⁢(X⁢kmodN+X⁢(−k)modN¯) X ep k 1 2 X k N X k N
12. i⁢ℑ⁢x⁢n x n X op ⁢k=12⁢(X⁢kmodN−X⁢(−k)modN¯) X op k 1 2 X k N X k N
13. x ep ⁢n=12⁢(x⁢nmodN+x⁢(−n)modN¯) x ep n 1 2 x n N x n N ℜ⁢X⁢k X k
14. x op ⁢n=12⁢(X⁢nmodN−x⁢(−n)modN¯) x op n 1 2 X n N x n N i⁢ℑ⁢X⁢k 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⁢(x⁢n+x⁢(−n)modN) x ep n 1 2 x n x n N ℜ⁢X⁢k X k
17. x op ⁢n=12⁢(x⁢n−x⁢(−n)modN) x op n 1 2 x n x n N ℑ⁢X⁢k X k

Note:

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

Complex Sinusoids in a nutshell

A complex sinusoid is denoted as ∀ n,k ,0≤n,k≤N−1:ei⁢2⁢πN⁢k⁢n 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 ≤2⁢π: ω k =2⁢πN⁢k 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

Figure 9: k=2 k 2 , N=16 N 16
Figure 9 (DFT9.png)
C sinusoid makes exactly kk revolutions of the circle and repeats when n=N n N at the n=0 n 0 spot.
Figure 10: N=16 N 16 , k=2 k 2 times around.
Figure 10 (DFT10.png)

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

Figure 11: k=2 k 2 , N=16 N 16 : note that there are 2 cycles
Figure 11 (DFT11.png)
Consider the following examples:

Example 4

Figure 12: N=8 N 8 , k=1 k 1 , ω k =2⁢π8⁢1=π4 ω k 2 8 1 4
Figure 12 (DFT12.png)

Example 5

Figure 13: N=8 N 8 , k=7 k 7 , ω k =2⁢π8⁢7=7⁢π4 ω k 2 8 7 7 4
Figure 13 (DFT13.png)
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.
Figure 14: ω k =−π4 ω k 4
Figure 14 (DFT14.png)

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 .

Figure 15
Figure 15 (DFT15.png)
In general, this is easy to show: ei⁢2⁢πN⁢k⁢n=ei⁢2⁢πN⁢k⁢n⁢ei⁢2⁢πN⁢N=ei⁢2⁢πN⁢(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

ei⁢2⁢πN⁢k⁢n=ei⁢ ω k ⁢n 2 N k n ω k n
(2)
ω k =2⁢πN⁢k ω k 2 N k
Figure 16
Figure 16 (DFT16.png)

Impact on DFT

Figure 17
Figure 17 (DFT17.png)
Matlab's fftshift command swaps left and right halfs of a vector:
Figure 18
Figure 18 (DFT18.png)

DFT Notation

x=B⁢α x B α α=BH⁢x α B x Now replace αα by capital XX: x=B⁢X x B X X=BH⁢x X B x x⁢n ↔ DFT X⁢k x n ↔ DFT X k

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