Title: Chapter 3. The Discrete-Time Fourier Analysis
1Chapter 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/
2Introduction
- 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.
3Introduction (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.
4The discrete-time Fourier transform (DTFT)
DTFT
IDTFT
Existence Condition x(n) is absolutely summable.
5DTFT
- 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
6Examples
- 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
7Two 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
8Symmetry
9Matlab 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
10Example
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)
11Examples
- 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.
12The 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.
13The 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.
14The properties of the DTFT
- 7. Convolution
- 8. Multiplication
- Periodic convolution (discussed in Chapter 5)
- 9. Energy
Parsevals Theorem
Energy density spectrum
15The 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
16Response 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.
18Response to sinusoidal sequences
Steady-state response
19Response to arbitrary sequences
- Condition
- Absolutely summable sequence
- LTI system
20Frequency 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
21Implementation in Matlab
22Sampling 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.
23Sampling
Continuous-time Fourier transform and inverse CTFT
- Absolutely integrable
- Omega is an analogy frequency in radians/sec
24Sampling
- 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
25The 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
26Meaning 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
27Ban-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
28Sampling 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.
29Reconstruction
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.
30Practical 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
31Zero-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.
32First-order-hold (FOH) interpolation
- In this case the adjacent samples are joined by
straight lines.
33Cubic-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
34Matlab 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)
35Reference 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