The Discrete Fourier Transform CS/BIOEN 4640: Image Processing - - PowerPoint PPT Presentation

The Discrete Fourier Transform

CS/BIOEN 4640: Image Processing Basics March 27, 2012

Review: Fourier Transform

Given a complex-valued function g : R → C, Fourier transform produces a function of frequency ω:

G(ω) = 1 √ 2π ∞

−∞

g(x) ·

• cos(ωx) − i · sin(ωx)
• dx

= 1 √ 2π ∞

−∞

g(x) · e−iωx dx

Review: The Dirac Delta

Definition

The Dirac delta or impulse is defined as

δ(x) = 0 for x = 0,

and

∞

−∞

δ(x) dx = 1

◮ The Dirac delta is not a function ◮ It is undefined at x = 0. ◮ Has the property

∞

−∞

f(x) δ(x) dx = f(0)

for any function f

The Fourier Transform of a Dirac Delta

F{δ(x)}(ω) = 1 √ 2π ∞

−∞

δ(x) · e−i ωxdx = 1 √ 2π e0 = 1 √ 2π

◮ In other words, Fourier of a Dirac is constant ◮ So, it has equal response at all frequencies

Convolution with a Dirac Delta

Convolving a function g(x) with a Dirac delta gives

(g ∗ δ)(x) = ∞

−∞

g(y) · δ(y − x) dy = g(x)

◮ So, convolving with Dirac is the identity operator ◮ Also can be seen in the Fourier domain:

F{g ∗ δ} = √ 2π F{g} · F{δ} = F{g}

The Comb

Definition

The comb function or Shah function is defined as an infinite sum of Dirac deltas:

III(x) =

∞

• k=−∞

δ(x − k)

III(x)

• −5

−4 −3 −2 −1 1 2 3 4 5 −0.5 0.5 1.0 1.5

Notice that just like the Dirac delta, the comb function is not a function

Using the Comb to Sample

Given a continuous function g(x), we can “sample” this function by multiplication by a comb:

¯ g(x) = g(x) · III(x) =

∞

• k=−∞

g(k) · δ(x − k)

Notice that ¯

g(x) is also not a function.

Using the Comb to Sample

The Discrete Fourier Transform

For a discrete signal g(u), where u = 0, 1, . . . , M, the discrete Fourier transform is given by

G(m) = 1 √ M

M−1

• u=0

g(u) ·

• cos
• 2πmu

M

• − i · sin
• 2πmu

M

• =

1 √ M

M−1

• u=0

g(u) · e−i 2π mu

M

Comparing Discrete and Continuous

Continuous Fourier Transform:

G(ω) = 1 √ 2π ∞

−∞

g(x) · e−iωx dx

Discrete Fourier Transform:

G(m) = 1 √ M

M−1

• u=0

g(u) · e−i 2π mu

M

Inverse Discrete Fourier Transform

The inverse DFT, analagous to the continuous case, just changes the sign in the exponent:

g(u) = 1 √ M

M−1

• u=0

G(m) · e i 2π mu

M

The Fourier Transform of the Comb

Fourier transform of a comb is another comb:

F {III(x)} = III ω 2π

The Fourier Transform of the Comb

Spacing of comb in time domain is inversely related to spacing in frequency domain:

F

• III

x τ

• = τ III

τω 2π

Sampled Signals in the Fourier Domain

If ¯

g(x) = g(x) · III(x), then ¯ G(ω) = √ 2π G(ω) ∗ III ω 2π

• ◮ We know convolution with δ is the identity

◮ So, this produces copies of the spectrum G shifted

to each peak of the comb.

Aliasing Caused by Sampling

The Nyquist-Shannon Sampling Theorem

Theorem

A bandlimited continuous signal can be perfectly reconstructed from a set of uniformly-spaced samples if the sampling frequency is twice the bandwidth of the signal.