Signal Processing and Linear Systems1

Nicholas Dworkwww.stanford.edu/~ndwork

Lecture 9: Properties of the DFT

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OutlineBackground

Theorems of the DFT

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kak2 =q

a21 + a22 + · · ·+ a2n

L2 Norm

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Kronecker Delta Vector

� = (1, 0, 0, . . . , 0)

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Matrix Transpose2

6664

a11 a12 · · · a1Na21 a22 a2N...

. . ....

aM1 aM2 aMN

3

7775

T

=

2

6664

a11 a21 · · · aM1a12 a22 aM2...

. . ....

a1N a2N aMN

3

7775

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Matrix Conjugate-Transpose2

6664

a11 a12 · · · a1Na21 a22 a2N...

. . ....

aM1 aM2 aMN

3

7775

⇤

=

2

6664

a11 a21 · · · aM1a12 a22 aM2...

. . ....

a1N a2N aMN

3

7775

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Suppose . Then the circular convolution of f and g is

Circular Convolution

f ~ g =NX

i=1

fi ⌧i{g}

f, g 2 CN

(f ~ g)[n] =NX

m=1

f [m] g [(n�m)modN ]

where is the circular shift operator of elements.⌧i i

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Recall: for a discrete LSI system S

S{x} =NX

i=1

xi ⌧i{h} = x~ h.

That is, the output equals the input circularly convolved with the impulse response.

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Discrete Fourier TransformAnalysis Equations:

Synthesis Equations:

Note: both the DFT and the IDFT can be done by a computer.

DFT{f}[m] =N�1X

n=0

f [n] exp⇣�i 2⇡mn

N

⌘

IDFT{F }[n] = 1N

N�1X

m=0

F [m] exp⇣i 2⇡

mn

N

⌘

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OutlineBackground

Theorems of the DFT

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Proof:

= exp

✓�i 2⇡ 0m

N

◆= 1.

DFT{�}[m] =N�1X

n=0

� exp⇣�i 2⇡ mn

N

⌘

DFT{�} = 1

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The DFT Matrix

The Discrete Fourier Transform is a linear transformation from a finite dimensional vector space to a finite dimensional vector space.

DFT : CN ! CN

Therefore, it can be represented by a matrix: W.

Similarly, the Inverse Discrete Fourier Transform can be represented by:

W�1 =1

NW ⇤

DFT{f} = W f

W�1

IDFT{F } = W�1 F = 1N

W ⇤ F

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The Unitary DFT Matrix

W�1 =1

NW ⇤

Let WU =1pN

W.

Then W�1U = W⇤U .

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Claim: if is a unitary matrix, then

Parseval’s Theorem

U kU xk2 = kxk2.

Proof:kU xk2 = kxk2 , kU xk22 = kxk22.

= x⇤U⇤Ux= x⇤Ix= x⇤x = kxk22.

Corollary: kDFT{x}k2 =pN kxk2.

kU xk22 = (U x)⇤(Ux)

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Convolution Theorem

DFT{f ~ g} = DFT{f}�DFT{g}

DFT{f � g} = DFT{f}~DFT{g}

Proof: see homework.

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For an LSI system, y = S{x},

The DFT of the output equals the DFT of the input times the transfer function (the DFT of the impulse response).

Y [m] = X[m]H[m]

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Shift Theorem

Proof:

DFT{⌧� f}[m] =N�1X

n=0

f [n��] exp⇣�i 2⇡mn

N

⌘

= exp

✓�i 2⇡m�

N

◆DFT{f}[m].

= exp

✓�i 2⇡m�

N

◆N�1X

ñ=0

f [ñ] exp

✓�i 2⇡mñ

N

◆

DFT{⌧� f}[m] = exp✓�i 2⇡m�

N

◆DFT{f}[m]

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Periodicity of the DFTSo far, we’ve been considering the DFT as a function that maps to .CNCN

But there’s nothing stopping us from computing the value of the DFT for frequencies outside of m in {0, …, N-1}.

When we do that (when we think of the domain of the DFT as all integers), we see that the DFT is periodic with fundamental period N.

So we can use any N consecutive values to reconstruct the input vector.

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Symmetry of the DFT

DFT{ real and even } = real and even

DFT{ real and odd } = imaginary and odd

DFT{ real } = Hermitian

Note: even and odd are defined according to the periodic extensions of the vectors.

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