Concepts in Biochemistry 3/e

1
Fourier transform
2
Response of Linear Shift-Invariant System
h(t)
h(t- ?)
Input
Output
?
t
?
t
Input
Output
t
t
3
Mathematical tool for Signal Analysis of Linear
System

• The mathematical tool that relates the
  frequency-domain description of the signal to its
  time-domain description is the Fourier transform.
• The continuous Fourier transform, or the Fourier
  transform (FT) for short.
• The discrete Fourier transform, or DFT for short.

4
Temporal Fourier Transform
5
02_01
Sketch of the interplay between the synthesis and
analysis equations embodied in Fourier
transformation.
6
Properties of The Fourier Transform
7
Properties of The Fourier Transform
8
(No Transcript)
9
(No Transcript)
10
Properties of The Fourier Transform
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
Properties of The Fourier Transform
16
Properties of The Fourier Transform
17
(No Transcript)
18
Dirac Delta Function
Dirac delta function definition
G(f)
g(t)
1
?(t)
f
t
0
19
Dirac delta function properties
20
Fourier transforms of Periodic Signals
21
02_19
Dirac Comb Function
g(t)
G(f)
22
Example
23
(No Transcript)
24
(No Transcript)
25
(No Transcript)
26
(No Transcript)
27
Problem 1
G(f)sinc(f)
f
28
Problem 2
t
G(f)sinc2(f)
f
29
3. Sinc pulse function
Problem 3
30
4. Radio frequency (RF) pulse function
Problem 4
31
Problem 5
t
-3
-2
-1
2
3
4
0
1
t
-3
-2
-1
2
3
0
1
32
Problem 6
33
Function
Fourier Transform Table
G(f)
g(t)
34
Function
G(f)
g(t)
35
The Inverse Relationship Between Time and
Frequency

• ?The time-domain and frequency-domain
  descriptions of a signal are inversely
• related to each other.
• ? A signal cannot be strictly limited in both
  time and frequency.
• ? The bandwidth of a signal provides a measure of
  the extent of the significant
• spectral content of the signal for positive
  frequencies.

? When the spectrum of a signal is symmetric with
a main lobe bounded by well- defined nulls
(i.e., frequencies at which the spectrum is
zero). ? A rectangular pulse of duration T
seconds has a main spectral lobe of total width
(2/T) hertz centered at the origin. ? The
bandwidth of this rectangular pulse is (1/T)
hertz. ? The definition of bandwidth is called
the null-to-null bandwidth.
36

• Another popular definition of bandwidth is the
  3-db bandwidth.
• Yet another measure for the bandwidth of a signal
  is the root mean-square (rms) bandwidth.
• The rms bandwidth of a low-pass signal g(t) with
  Fourier transform G(f)

• Time-Bandwidth Product
• (duration) (bandwidth) constant
• The rms duration of the signal g(t) is

37
Numerical Computation of the Fourier Transform

• The fast Fourier transform algorithm is itself
  derived from the discrete Fourier
• transform, both time and frequency are
  represented in discrete form.
• The discrete Fourier transform provides an
  approximation to the Fourier
• transform.
• If the samples are uniform spaced by Ts seconds,
  the spectrum of the signal
• becomes periodic, repeating every fs1/Ts Hz. Let
  N denote the number of
• frequency samples contained in an interval fs.
• Hence, the frequency resolution involved in the
  numerical computation of the
• Fourier transform is defined by

where T NTs is the total duration of the
signal.
38
Interpretations of the DFT and the IDFT

• The discrete Fourier transform (DFT) of the
  sequence gn

Consider then a finite data sequence g0,
g1,..,,gN-1 which can represent the result of
sampling an analog signal g(t) at times t0, Ts,
(N-1)Ts, where Ts is the sampling interval.

• The sequence G0, G1,GN-1 is called the
  transform sequence. The inverse discrete Fourier
• transform (IDFT) of Gk

39
02_34
The DFT as an analyzer of the data sequence gn
40
02_34
The IDFT as a synthesizer of gn
41
Fast Fourier Transform Algorithms
FFT algorithms are computationally efficient
because they use a greatly reduced number of
arithmetic operations as compared to brute force
computation of the DFT. An FFT algorithm attains
its computational efficiency by following a
divide-and-conquer strategy, whereby the original
DFT computation is decomposed successively into
smaller DFT computations.
42
The coefficient Wn multiplying yn is called a
twiddle factor.
43
02_35
Computation for The Data Sequence
A signal-flow graph consists of an
interconnection of nodes and branches. The
direction of signal transmission along a branch
is indicated by an arrow. A branch multiplies
the variable at a node by the branch
transmittance.
. 4-point DFT into two 2-point DFTs
Trivial case of 2-point DFT
8-point DFT into two 4-point DFTs
44
02_36
Decimation-in-frequency FFT algorithm.
45

• The FFT algorithm is referred to as a
  decimation-in-frequency algorithm, because the
  transform (frequency) sequence Gk, is divided
  successively into smaller subsequences. In
  another popular FFT algorithm, called a
  decimation-in-time algorithm, the data (time)
  sequence gn is divided successively into smaller
  subsequences.

P.88
46
02_02tbl
P.88