Chapter 3. The Discrete-Time Fourier Analysis

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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/

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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.

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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.

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The discrete-time Fourier transform (DTFT)
DTFT
IDTFT
Existence Condition x(n) is absolutely summable.
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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

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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

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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

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Symmetry
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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

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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)
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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.

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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.

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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.

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The properties of the DTFT

• 7. Convolution
• 8. Multiplication
• Periodic convolution (discussed in Chapter 5)
• 9. Energy

Parsevals Theorem
Energy density spectrum
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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

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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
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• 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.

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Response to sinusoidal sequences
Steady-state response
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Response to arbitrary sequences

• Condition
• Absolutely summable sequence
• LTI system

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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

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Implementation in Matlab
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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.

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Sampling
Continuous-time Fourier transform and inverse CTFT

1. Absolutely integrable
2. Omega is an analogy frequency in radians/sec

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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
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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

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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

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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

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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.

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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.
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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

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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.

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First-order-hold (FOH) interpolation

• In this case the adjacent samples are joined by
  straight lines.

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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

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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)
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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