Homework: 7.21, 7.23 Tutorial Problems: 7.25, 7.37, 7 7a.pdf · 2020. 11. 23. · Homework: 7.21, 7.23 Tutorial Problems: 7.25, 7.37, 7.40 Signals & SystemsSampling P1. Chapter 7:

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


Introduction


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


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


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 )


Sampling discards most of points in the original signals.Is there any information loss in sampling?


Observation (1/2)


Observation (2/2)

Sampling: the frequency should be high enough

Reconstruction: the interpolation should be smooth enough


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)


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


Reconstruction Problem

Given the sampled signal, can we perfectly reconstruct the signal beforesampling?


Reconstruction (1/2)

Scenario of ωs > 2ωM


Reconstruction (2/2)Scenario of ωs ≤ 2ωM


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


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


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?


Interpretation of Zero-Order Hold

Zero-order hold: sampling + interpolation with rectangular impulse response

An approximation of the signal to be sampled.


Frequency Analysis (1/2)Step 1: Impulse-train sampling

Step 2: Frequency response of h0(t)


Frequency Analysis (2/2)


Reconstruction

H(jω) should be a idea low-pass filter from −ωs/2 to ωs/2


First-Order Hold

First-order hold: sampling + interpolation with triangular wave

How to reconstruct?


Summary: Sampling Approaches


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.


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.


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?


Homework: 7.21, 7.23 Tutorial Problems: 7.25, 7.37, 7 7a.pdf · 2020. 11. 23. · Homework: 7.21, 7.23 Tutorial Problems: 7.25, 7.37, 7.40 Signals & SystemsSampling P1. Chapter 7:

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