Homework: 7.21, 7.23Tutorial Problems: 7.25, 7.37, 7.40
Signals & Systems Sampling P1
Chapter 7: Sampling
Department of Electrical & Electronic EngineeringSouthern Univerisity of Science & Technology
2020 - FallLast Update on: Tuesday 24th November, 2020
Signals & Systems Sampling P2
Introduction
Signals & Systems Sampling P3
Video RecordingSignal to be sampled: real scene (continuous-time signals)
Sampling: record by camera with a rate of 24, 25 or 30 frames per second
Sampled signal: video tapes, mp4 files, avi files and etc. (discrete-timesignals)
Reconstruction: watch via eyes and interpret in the brain
In our consciousness, the real scene can be reconstructed without informationloss
Signals & Systems Sampling P4
Outline
Sampling is a general procedure to generate DT signals from CT signals,where information of the original signals can be kept
Core sampling theory:I Impulse train, zero-order hold, first-order hold and etcI Analysis in frequency domainI Nyquist rate
Undersampling: Aliasing
Application: process continuous-time signals discretely
More sampling techniques: decimation, downsampling and upsampling
Signals & Systems Sampling P5
Impulse-Train SamplingMathematically, sampling can be represented by multiplication
Sampling function: p(t) =∑∞
n=−∞ δ(t − nT )Sampling period: T
Sampling:
xp(t) = x(t)× p(t) = x(t)×∞∑
n=−∞δ(t − nT ) =
∞∑n=−∞
x(nT )δ(t − nT )
Signals & Systems Sampling P6
Sampling discards most of points in the original signals.Is there any information loss in sampling?
Signals & Systems Sampling P7
Observation (1/2)
Signals & Systems Sampling P8
Observation (2/2)
Sampling: the frequency should be high enough
Reconstruction: the interpolation should be smooth enough
Signals & Systems Sampling P9
Frequency Analysis (1/2)Theoretical tool: continuous-time Fourier transform
Principle:
x(t)× p(t) � 12π
X (jω) ∗ P(jω)
Fourier series of p(t):
ak =1
T
∫ T/2−T/2
∞∑n=−∞
δ(t − nT )e−jkωs tdt where ωs =2π
T
=1
T
∫ T/2−T/2
δ(t)e−jkωs tdt =1
T
Fourier Transform of p(t):
P(jω) = 2πak
+∞∑k=−∞
δ(ω − kωs) =2π
T
+∞∑k=−∞
δ(ω − kωs)
Signals & Systems Sampling P10
Frequency Analysis (2/2)Fourier transform of sampled signal xp(t):
Xp(jω) =1
2π
∫ ∞−∞
X (jθ)P(j(ω − θ))dθ
=1
T
∞∑k=−∞
X (j(ω − kωs))
Sampling: the Fourier transform of input signal is repeated with period ωs
Signals & Systems Sampling P11
Reconstruction Problem
Given the sampled signal, can we perfectly reconstruct the signal beforesampling?
Signals & Systems Sampling P12
Reconstruction (1/2)
Scenario of ωs > 2ωM
Signals & Systems Sampling P13
Reconstruction (2/2)Scenario of ωs ≤ 2ωM
Signals & Systems Sampling P14
Sampling Theorem
Let x(t) be a band-limited signal with
X (jω) = 0 for |ω| > ωM .
Then, x(t) is uniquely determined by its samples x(nT ) or xp(t) if
ωs =2π
T> 2ωM ,
where 2ωM is referred to as the Nyquist rate.
Questions:I How about ωs = 2ωM?I Sampling on band-pass signals
Signals & Systems Sampling P15
Signal Reconstruction: Interpolation
If ωs > 2ωM , original signal can be perfectly reconstructed by ideal low-passfilter.
Time domain interpretation of lowpass filtering
xr (t) = xp(t) ∗ h(t) =+∞∑
n=−∞x(nT )h(t − nT )
=+∞∑
n=−∞x(nT )
sin ωs2 (t − nT )ωs2 (t − nT )
=+∞∑
n=−∞x(nT )sinc(
t − nTT
)
Ideal lowpass filtering: interpolation with sinc function
Signals & Systems Sampling P16
Zero-Order HoldIt’s difficult to generate ideal impulse chain in practical implementation.
xp(t) =∞∑
n=−∞x(nT )δ(t − nT )
Alternative approach: zero-order hold
How to interpret the system of ”zero-order hold” mathematically?
Signals & Systems Sampling P17
Interpretation of Zero-Order Hold
Zero-order hold: sampling + interpolation with rectangular impulse response
An approximation of the signal to be sampled.
Signals & Systems Sampling P18
Frequency Analysis (1/2)Step 1: Impulse-train sampling
Step 2: Frequency response of h0(t)
Signals & Systems Sampling P19
Frequency Analysis (2/2)
Signals & Systems Sampling P20
Reconstruction
H(jω) should be a idea low-pass filter from −ωs/2 to ωs/2
Signals & Systems Sampling P21
First-Order Hold
First-order hold: sampling + interpolation with triangular wave
How to reconstruct?
Signals & Systems Sampling P22
Summary: Sampling Approaches
Signals & Systems Sampling P23
Problem 1
Problem (7.5)
Let x(t) be a signal with Nyquist rate ω0. Also, let
y(t) = x(t)p(t − 1),
where
p(t) =∞∑
n=−∞δ(t − nT ), and T < 2π
ω0.
Specify the constraints on the magnitude and phase of the frequency response ofa filter that gives x(t) as its output when y(t) is the input.
Signals & Systems Sampling P24
Problem 2
Problem (7.7)
A signal x(t) undergoes a zero-order hold operation with an effective samplingperiod T to produce a signal x0(t). Let x1(t) denote the result of a first-orderhold operation on x(t). Specify the frequency response of a filter that producesx1(t) as its output when x0(t) is the input.
Signals & Systems Sampling P25
Problem 3
Problem (7.36)
Let x(t) be a band-limited signal such that X (jω) = 0 for |ω| ≥ π/T .(a) If x(t) is sampled using a sampling period T , determine an interpolatingfunction g(t) such that
dx(t)
dt=
∞∑n=−∞
x(nT )g(t − nT ).
(b) Is the function g(t) unique?
Signals & Systems Sampling P26
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