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Discrete Fourier transform, properties , formulas and concepts
Typology: Lecture notes
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Recall the DTFT:
X(ω) =
n=−∞
x(n)e−jωn.
DTFT is not suitable for DSP applications because
A finite signal measured at N points:
x(n) =
0 , n < 0 , y(n), 0 ≤ n ≤ (N − 1), 0 , n ≥ N,
where y(n) are the measurements taken at N points.
Sample the spectrum X(ω) in frequency so that
X(k) = X(k∆ω), ∆ω =
2 π N
X(k) =
n=
x(n)e−j^2 π^
knN DFT.
The inverse DFT is given by:
x(n) =
k=
X(k)ej^2 π^
knN .
x(n) =
k=
m=
x(m)e−j^2 π^
kmN
ej^2 π^
knN
m=
x(m)
k=
e−j^2 π^
k(m−n) N
δ(m−n)
= x(n).
X(k + N ) =
n=
x(n)e−j^2 π^
(k+N )n N
n=
x(n)e−j^2 π^
knN
e−j^2 πn
= X(k)e−j^2 πn^ = X(k) =⇒
the DFT spectrum is periodic with period N (which is expected, since the DTFT spectrum is periodic as well, but with period 2 π).
Example: DFT of a rectangular pulse:
x(n) =
1 , 0 ≤ n ≤ (N − 1), 0 , otherwise.
X(k) =
n=
e−j^2 π^
knN = N δ(k) =⇒
the rectangular pulse is “interpreted” by the DFT as a spectral line at frequency ω = 0.
What happens with the DFT of this rectangular pulse if we increase N by zero padding:
{y(n)} = {x(0),... , x(M − 1), (^0) ︸ , 0 ,... ,︷︷ 0 ︸ N −M positions
where x(0) = · · · = x(M − 1) = 1. Hence, DFT is
Y (k) =
n=
y(n)e−j^2 π^
n=
y(n)e−j^2 π^
knN
sin(πkMN ) sin(π (^) Nk )
e−jπ^
k(M −1) N (^).
Remarks:
which follows easily by checking WH^ W = WWH^ = N I, where I denotes the identity matrix. Hermitian transpose:
xH^ = (xT^ )∗^ = [x(1)∗, x(2)∗,... , x(N )∗].
Also, “∗” denotes complex conjugation.
Frequency Interval/Resolution: DFT’s frequency resolution
Fres ∼
[Hz]
and covered frequency interval
∆F = N ∆Fres =
= Fs [Hz].
Frequency resolution is determined only by the length of the observation interval, whereas the frequency interval is determined by the length of sampling interval. Thus
Question: Does zero padding alter the frequency resolution?
Answer: No, because resolution is determined by the length of observation interval, and zero padding does not increase this length.
Example (DFT Resolution): Two complex exponentials with two close frequencies F 1 = 10 Hz and F 2 = 12 Hz sampled with the sampling interval T = 0. 02 seconds. Consider various data lengths N = 10, 15 , 30 , 100 with zero padding to 512 points.
DFT with N = 10 and zero padding to 512 points. Not resolved: F 2 − F 1 = 2 Hz < 1 /(N T ) = 5 Hz.
DFT with N = 100 and zero padding to 512 points. Resolved: F 2 − F 1 = 2 Hz > 1 /(N T ) =
Construct a periodic sequence by periodic repetition of x(n) every N samples:
{x˜(n)} = {... , x ︸ (0),... , x︷︷ (N − 1)︸
{x(n)}
, x ︸ (0),... , x︷︷ (N − 1)︸ {x(n)}
The discrete version of the Fourier Series can be written as
x ˜(n) =
k
Xkej^2 π^
k
X^ ˜(k)ej^2 π^ knN^ =^1 N
k
X^ ˜(k)W −kn,
where X˜(k) = N Xk. Note that, for integer values of m, we have
W −kn^ = ej^2 π^
knN = ej^2 π^
(k+mN )n N (^) = W −(k+mN^ )n.
As a result, the summation in the Discrete Fourier Series (DFS) should contain only N terms:
x ˜(n) =
k=
X^ ˜(k)ej^2 π^ knN^ DFS.
− DFT is applied to finite sequence x(n), − DFS is applied to periodic sequence x˜(n).
− CFS represents a continuous periodic signal using an infinite number of complex exponentials, whereas − DFS represents a discrete periodic signal using a finite number of complex exponentials.
Linearity
Circular shift of a sequence: if X(k) = DFT {x(n)} then
X(k)e−j^2 π^
kmN = DFT {x((n − m) mod N )}
Also if x(n) = DFT −^1 {X(k)} then
x((n − m) mod N ) = DFT −^1 {X(k)e−j^2 π^
kmN }
where the operation mod N denotes the periodic extension x ˜(n) of the signal x(n):
x ˜(n) = x(n mod N ).
= W km
n=
x((n − m)modN )W k(n−m)modN
= W kmX(k),
where we use the facts that W k(lmodN^ )^ = W kl^ and that the order of summation in DFT does not change its result.
Similarly, if X(k) = DFT {x(n)}, then
X((k − m)modN ) = DFT {x(n)ej^2 π^
mnN }.
DFT: Parseval’s Theorem
n=
x(n)y∗(n) =
k=
X(k)Y∗(k)
Using the matrix formulation of the DFT, we obtain
yH^ x =
N I
DFT: Circular Convolution
If X(k) = DFT {x(n)} and Y (k) = DFT {y(n)}, then
X(k)Y (k) = DFT {{x(n)} ~ {y(n)}}
Here, ~ stands for circular convolution defined by
{x(n)} ~ {y(n)} =
m=
x(m)y((n − m) mod N ).
DFT {{x(n)} ~ {y(n)}}
n=
m=0 x(m)y((n^ −^ m) mod^ N^ )
{x(n)}~{y(n)}
W kn
m=
N − 1 n=0 y((n^ −^ m) mod^ N^ )W^
kn
Y (k)W km
x(m)
= Y (k)
m=
x(m)W km ︸ ︷︷ ︸ X(k)
= X(k)Y (k).
