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Chapter 3. The Discrete-Time Fourier Analysis

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Title: Chapter 3. The Discrete-Time Fourier Analysis


1
Chapter 3. The Discrete-Time Fourier Analysis
  • Gao Xinbo
  • School of E.E., Xidian Univ.
  • Xbgao_at_ieee.org
  • http//see.xidian.edu.cn/teach/matlabdsp/

2
Introduction
  • A linear and time-invariant system can be
    represented using its response to the unit sample
    sequence.
  • h(n) is called as the unit impulse response
  • y(n)x(n)h(n) system response
  • The convolution representation is based on the
    fact that any signal can be represented by a
    linear combination of scaled and delayed unit
    samples.
  • We can also represent any arbitrary discrete
    signal as a linear combination of basis signals
    introduced in Chapter 2.

3
Introduction (cont)
  • Each basis signal set provides a new signal
    representation.
  • Each representation has some advantages and
    disadvantages depending upon the type of system
    under consideration.
  • When the system is linear and time-invariant,
    only one representation stands out as the most
    useful. It is based on the complex exponential
    signal set and is called the
    discrete-time Fourier Transform.

4
The discrete-time Fourier transform (DTFT)
DTFT
IDTFT
Existence Condition x(n) is absolutely summable.
5
DTFT
  • F. transforms a discrete signal x(n) into a
    complex-valued continuous function X of real
    variable w, called a digital frequency, which is
    measured in radians.
  • Time domain --? Frequency domain
  • Discrete --? Continuous
  • Real valued --? Complex-valued
  • Summation --? integral
  • The range of w
  • The integral range of w

6
Examples
  • 3.1 Determine the discrete-time Fourier transform
    of
  • Result visualization with Matlab
  • Complex function magnitude and angle
  • real/imaginary part with respect to w
  • The range of w
  • showing interval 0,pi

7
Two important properties
  • Periodicity
  • The DTFT is periodic in w with period 2pi
  • Implication we need only one period for analysis
    and not the whole domain
  • Symmetry
  • For real-valued x(n), X is conjugate symmetric.
  • Implication to plot X, we now need to consider
    only a half of X 0,pi

8
Symmetry
9
Matlab Implementation
  • If x(n) is of infinite duration, then Matlab can
    not be used directly to compute X from x(n).
  • We can use it to evaluate the expression X over
    0,pi frequencies and then plot its magnitude
    and angle (or real and imaginary parts).
  • For a finite duration, the DTFT can be
    implemented as a matrix-vector multiplication
    operation.
  • w continuous--?discrete

10
Example
  • x(n)1,2,3,4,5

n-13 x15 K0500 w(pi/500)k Xx(exp(-j
pi/500).(nk) magXabs(X) angXangle(X) realX
real(X) imagX imag(X) subplot(2,2,1)
plot(k/500,magX) grid xlabel(frequency in pi
unit) title(Magnitude Part)
11
Examples
  • Example3.5
  • Complex-valued signal
  • The DTFT is periodic in w but it is not
    conjugate-symmetric.
  • Example3.6
  • Real-valued signal
  • Periodic in w and conjugate-symmetric
  • Therefore for real sequence we will plot their
    Fourier transform magnitude and angle responses
    from 0 to pi.

12
The properties of the DTFT
  • 1. Linearity
  • The DTFT is a linear transformation.
  • 2. Time shifting
  • A shift in the time domain corresponds to the
    phase shifting.
  • 3. Frequency shifting
  • Multiplication by a complex exponential
    corresponds to a shift in the frequency domain.
  • 4. Conjugation
  • Conjugation in the time domain corresponds to the
    folding and conjugation in the frequency domain.

13
The properties of the DTFT
  • 5. Folding
  • Folding in the time domain corresponds to the
    folding in the frequency domain.
  • 6. Symmetries in real sequence
  • Implication If the sequence x(n) is real and
    even, then X is also real and even.

14
The properties of the DTFT
  • 7. Convolution
  • 8. Multiplication
  • Periodic convolution (discussed in Chapter 5)
  • 9. Energy

Parsevals Theorem
Energy density spectrum
15
The frequency domain representation of LTI system
  • The Fourier transform representation is the most
    useful signal representation for LTI systems.
  • It is due to the following result
  • Response to a complex exponential ejw0n
  • Response to sinusoidal sequences
  • Response to arbitrary sequences

16
Response to a complex exponential
  • Frequency response the DTFT of an impulse
    response is called the frequency response (or
    transfer function) of an LTI system and is
    denoted by H().

The output sequence is the input exponential
sequence modified by the response of the system
at frequency w0
17
  • In general, the frequency response H is a complex
    function of w.
  • The magnitude H is called the magnitude (gain)
    response function, and
  • the the angle is called the phase response
    function.

18
Response to sinusoidal sequences
Steady-state response
19
Response to arbitrary sequences
  • Condition
  • Absolutely summable sequence
  • LTI system

20
Frequency response function from difference
equations
  • When an LTI system is represented by the
    difference equation,
  • then to evaluate its frequency response , we
    would need the impulse response h(n).
  • We know that when , then y(n)
    must be

21
Implementation in Matlab
22
Sampling and reconstruction of analog signals
  • Analogy signals can be converted into discrete
    signals using sampling and quantization
    operations analogy-to-digital conversion, or ADC
  • Digital signals can be converted into analog
    signals using a reconstruction operation
    digital-to-analogy conversion, or DAC
  • Using Fourier analysis, we can describe the
    sampling operation from the frequency-domain
    view-point, analyze its effects and then address
    the reconstruction operation.
  • We will also assume that the number of
    quantization levels is sufficiently large that
    the effect of quantization on discrete signals is
    negligible.

23
Sampling
Continuous-time Fourier transform and inverse CTFT
  1. Absolutely integrable
  2. Omega is an analogy frequency in radians/sec

24
Sampling
  • Sample xa(t) at sampling interval Ts sec apart to
    obtain the discrete-time signal x(n)

1. X is a countable sum of amplitude-scaled,
frequency-scaled, and translated version of Xa 2.
The above relation is known as the aliasing
formula
25
The analog and digital frequencies
Fs the sampling frequency, sam/sec
  • Amplitude scaled factor 1/Ts
  • Frequency-scaled factor ?OTs (?02p)
  • Frequency-translated factor 2pk/Ts

26
Meaning of Fig. 3.10
  • Suppose signal band is limited to ?0,
  • If Ts is small, ?0Tsltp, or
  • F0 ?0/2 p lt Fs/21/2Ts
  • Then the freq. Resp. of x(t) is an infinite
    replica series of its analog signal xa(t),
  • If Ts is large, ?0Tsgtp, or
  • F0 ?0/2 p gt Fs/21/2Ts
  • Then the freq. Resp. of x(t) is a overlaped
    replica of its analog signal xa(t), so cannot be
    reconstructed

27
Ban-limited signal
  • A signal is band-limited if there exists a finite
    radians frequency ?0 such that Xa(j ?) is zero
    for ? gt ?0.
  • The frequency F0 ?0 /2pi is called the signal
    bandwidth in Hz
  • Referring to Fig.3.10, if pigt ?0Ts, then

28
Sampling Principle
  • A band-limited signal xa(t) with bandwidth F0 can
    be reconstructed from its sample values
    x(n)xa(nTs) if the sampling frequency Fs1/Ts is
    greater than twice the bandwidth F0 of xa(t) , Fs
    gt2 F0.
  • Otherwise aliasing would result in x(n). The
    sampling rate of 2 F0 for an analog band-limited
    signal is called the Nyquist rate.

29
Reconstruction
Interpolating formula
1. Lowpass filter band-limited to the
-Fs/2,Fs/2 band 2. The ideal interpolation is
not practically feasible because the entire
system is noncausal and hence not realizable.
30
Practical D/A converters
  • Zero-order-hold (ZOH) interpolation
  • In this interpolation a given sample value is
    held for the sample interval until the next
    sample is received.
  • It can be obtained by filtering the impulse train
    through an interpolating filter of the form

31
Zero-order-hold (ZOH) interpolation
  • The resulting signal is a piecewise-constant
    (staircase) waveform which requires an
    appropriately designed analog post-filter for
    accurate waveform reconstruction.

32
First-order-hold (FOH) interpolation
  • In this case the adjacent samples are joined by
    straight lines.

33
Cubic-order-hold (COH) interpolation
  • This approach uses spline interpolation for a
    smoother, but not necessarily more accurate,
    estimate of the analog signal between samples.
  • Hence this interpolation does not require an
    analog post-filter
  • The smoother reconstruction is obtained by using
    a set of piecewise continuous third-order
    polynomials called cubic splines

34
Matlab Implementation
Nn1n2 tt1t2 Fs1/Ts nTs nTs Xa
xsinc(Fs(ones(length(n),1)t-
nTsones(1,length(t))))
ZOH stairs FOH plot Cubic spline interpolation
Xaspline(nTs, x, t)
35
Reference and assignments
  • Textbook pp40 to pp73
  • Chinese reference book pp2834,3743
  • pp1925
  • Exercises
  • 1 Textbook p3.1 p3.3b,e p3.4
  • 2 Textbook 3.16b3.18 Optional p3.7 3.15
  • 3 Optional Textbook p3.20a,b
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