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 :
Z ∞
h
i
1
G(ω) = √
g(x) · cos(ωx) − i · sin(ωx) dx
2π Z−∞
∞
1
g(x) · e−iωx dx
=√
2π −∞
Review: The Dirac Delta
Definition
The Dirac delta or impulse is defined as
Z
δ(x) = 0 for x 6= 0,
∞
δ(x) dx = 1
and
−∞
I
The Dirac delta is not a function
I
It is undefined at x = 0.
I
Has the property
Z
∞
f (x) δ(x) dx = f (0)
−∞
for any function f
The Fourier Transform of a Dirac Delta
Z ∞
1
F{δ(x)}(ω) = √
δ(x) · e−i ωx dx
2π −∞
1
= √ e0
2π
1
=√
2π
I
In other words, Fourier of a Dirac is constant
I
So, it has equal response at all frequencies
Convolution with a Dirac Delta
Convolving a function g(x) with a Dirac delta gives
Z
∞
(g ∗ δ)(x) =
g(y) · δ(y − x) dy
−∞
= g(x)
I
So, convolving with Dirac is the identity operator
I
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:
∞
X
III(x) =
δ(x − k)
k=−∞
1.5
●
●
●
III(x)
●
●
1.0
●
●
●
●
●
●
Notice that just like the
Dirac delta, the comb
function is not a function
0.5
−5
−4
−3
−2
−1
−0.5
0
1
2
3
4
5
Using the Comb to Sample
Given a continuous function g(x), we can “sample” this
function by multiplication by a comb:
ḡ(x) = g(x) · III(x)
∞
X
=
g(k) · δ(x − k)
k=−∞
Notice that ḡ(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
M−1
h mu mu i
1 X
G(m) = √
g(u) · cos 2π
− i · sin 2π
M
M
M u=0
M−1
mu
1 X
=√
g(u) · e−i 2π M
M u=0
Comparing Discrete and Continuous
Continuous Fourier Transform:
1
G(ω) = √
2π
Z
∞
g(x) · e−iωx dx
−∞
Discrete Fourier Transform:
M−1
mu
1 X
G(m) = √
g(u) · e−i 2π M
M u=0
Inverse Discrete Fourier Transform
The inverse DFT, analagous to the continuous case, just
changes the sign in the exponent:
M−1
mu
1 X
g(u) = √
G(m) · e i 2π M
M u=0
The Fourier Transform of the Comb
Fourier transform of a comb is another comb:
ω
F {III(x)} = III
2π
The Fourier Transform of the Comb
Spacing of comb in time domain is inversely related to
spacing in frequency domain:
n x o
τ ω F III
= τ III
τ
2π
Sampled Signals in the Fourier Domain
If ḡ(x) = g(x) · III(x), then
√
Ḡ(ω) =
ω
2π G(ω) ∗ III
2π
I
We know convolution with δ is the identity
I
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.