Transcript Chapter2_Lect8.ppt

Chapter 2
Discrete Fourier Transform (DFT)
Topics:
 Discrete Fourier Transform.
•
•
•
Using the DFT to Compute the Continuous Fourier
Transform.
Comparing DFT and CFT
Using the DFT to Compute the Fourier Series
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
Eeng 360 1
Discrete Fourier Transform (DFT)
Definition: The Discrete Fourier Transform (DFT) is defined by:
Where n = 0, 1, 2, …., N-1
The Inverse Discrete Fourier Transform (IDFT) is defined by:
where k = 0, 1, 2, …., N-1.
The Fast Fourier Transform (FFT) is a fast algorithm for evaluating
the DFT.
Eeng 360 2
Using the DFT to Compute the Continuous Fourier Transform
 Suppose the CFT of a waveform w(t) is to be evaluated using DFT.
1.
The time waveform is first windowed (truncated) over the interval (0, T) so
that only a finite number of samples, N, are needed. The windowed waveform
ww(t) is
2.
The Fourier transform of the windowed waveform is
3.
Now we approximate the CFT by using a finite series to represent the integral,
t = k∆t, f = n/T, dt = ∆t, and ∆t = T/N
Eeng 360 3
Computing CFT Using DFT
• We obtain the relation between the CFT and DFT; that is,
f = n/T and ∆t = T/N
• The sample values used in the DFT computation are x(k) = w(k∆t),
• If the spectrum is desired for negative frequencies
– the computer returns X(n) for the positive n values of 0,1, …, N-1
– It must be modified to give spectral values over the entire
fundamental range of -fs/2 < f <fs/2.
For positive frequencies we use
For Negative Frequencies
Eeng 360 4
Comparison of DFT and
the Continuous Fourier
Transform (CFT)
Relationship between the DFT
and the CFT involves three
concepts:
• Windowing,
• Sampling,
• Periodic sample generation
Eeng 360 5
Comparison of DFT and
the Continuous Fourier
Transform (CFT)
Relationship between the DFT
and the CFT involves three
concepts:
• Windowing,
• Sampling,
• Periodic sample generation
Eeng 360 6
Fast Fourier Transform
 The Fast Fourier Transform (FFT) is a fast algorithm for evaluating DFT.
Block diagrams depicting the
decomposition of an inverse DTFS
as a combination of lower order
inverse DTFS’s.
(a) Eight-point inverse DTFS
represented in terms of two fourpoint inverse DTFS’s.
(b) four-point inverse DTFS
represented in terms of two-point
inverse DTFS’s.
(c) Two-point inverse DTFS.
Eeng 360 7
Using the DFT to Compute the Fourier Series
 The Discrete Fourier Transform (DFT) may also be used to compute the
complex Fourier series.
 Fourier series coefficients are related to DFT by,
1
cn  X (n)
N
1
cn  X (n),
N
1
cn  X ( N  n),
N
N
0n
2
N
- n0
2
 Block diagram depicting the sequence of operations involved in approximating the FT
with the DTFS.
Eeng 360 8
Ex. 2.17 Use DFT to compute the spectrum of a Sinusoid
Eeng 360 9
Ex. 2.17 Use DFT to compute the spectrum of a Sinusoid
Spectrum of a sinusoid obtained by using the MATLAB DFT.Eeng 360
10
Using the DFT to
Compute the
Fourier Series
The DTFT and length-N DTFS of a
32-point cosine.
The dashed line denotes the CFT.
While the stems represent N|X[k]|.
(a) N = 32
(b) N = 60
(c) N = 120.
Eeng 360 11
Using the DFT
to Compute the
Fourier Series
The DTFS approximation to the FT of x(t) = cos(2(0.4)t) + cos(2(0.45)t). The stems
denote |Y[k]|, while the solid lines denote CFT. (a) M = 40. (b) M = 2000. (c) Behavior in
the vicinity of the sinusoidal frequencies for M = 2000. (d) Behavior in the vicinity of the
sinusoidal frequencies for M = 2010
Eeng 360 12