Bài giảng Digital signal processing - Chapter 1: Sampling and Reconstruction

Chapter 1

Sampling and Reconstruction

Nguyen Thanh Tuan,_CM_lic._kE_ton_eg_d._{it Master subtitle style}

Department of Telecommunications (113B3)

Ho Chi Minh City University of Technology

Email: [email protected]om

Content

 Sampling

 Sampling theorem

 Spectrum of sampling signals

 Anti-aliasing pre-filter

 Ideal pre-filter

 Practical pre-filter

 Analog reconstruction

 Ideal reconstructor

 Practical reconstructor

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1. Introduction

 A typical signal processing system includes 3 stages:

 The analog signal is digitalized by an A/D converter

 The digitalized samples are processed by a digital signal processor.

 The digital processor can be programmed to perform signal processing

operations such as filtering, spectrum estimation. Digital signal processor can be

a general purpose computer, DSP chip or other digital hardware.

 The resulting output samples are converted back into analog by a

D/A converter.

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2. Analog to digital conversion

 Analog to digital (A/D) conversion is a three-step process.

x_Q(n)

x(nT)≡x(n)

11010

x(t)

Sampler

t=nT

Quantizer

Coder

A/D converter

x_Q(n)

x(t)

x(n)

111

110

101

100

011

010

001

000

n

t

n

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3. Sampling

 Sampling is to convert a continuous time signal into a discrete time

signal. The analog signal is periodically measured at every T seconds

 x(n)≡x(nT)=x(t=nT), n=…-2, -1, 0, 1, 2, 3…

 T: sampling interval or sampling period (second);

?

 Fs=1/T: sampling rate or frequency (samples/second or Hz)

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Example 1

 The analog signal x(t)=2cos(2πt) with t(s) is sampled at the rate F_s=4

Hz. Find the discrete-time signal x(n) ?

Solution:

 x(n)≡x(nT)=x(n/Fs)=2cos(2πn/Fs)=2cos(2πn/4)=2cos(πn/2)

n

0

1

2

3

4

x(n)

2

0

-2

0

2

 Plot the signal

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Sampling and Reconstruction

Example 2

 Consider the two analog sinusoidal signals

7

x₁(t)  2cos(2 t),

8

1

x₂(t)  2cos(2 t); t(s)

8

These signals are sampled at the sampling frequency F_s=1 Hz.

Find the discrete-time signals ?

Solution:

1

7 1

7

x₁(n)  x₁(nT)  x₁(n )  2cos(2 n)  2cos( n)

F

8 1

4

s

1



4

 2cos((2 )n)  2cos( n)

4

1

11

x₂(n)  x₂(nT)  x₂(n )  2cos(2 n)  2cos( n)

81

1

F

4

s

 Observation: x₁(n)=x₂(n)  based on the discrete-time signals, we

cannot tell which of two signals are sampled ? These signals are

called “alias”

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Sampling and Reconstruction

F₂=1/8 Hz

F₁=7/8 Hz

F_s=1 Hz

Fig: Illustration of aliasing

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Sampling and Reconstruction

4. Aliasing of Sinusoids

 In general, the sampling of a continuous-time sinusoidal signal

at a sampling rate F =1/T results in a

x(t)  Acos(2F t )

s

0

discrete-time signal x(n).

x_k(t)  Acos(2F t )

 The sinusoids

is sampled at F_s, resulting

k

in a discrete time signal x_k(n).

 If F_k=F₀+kF_s, k=0, ±1, ±2, …., then x(n)=x_k(n) .

Proof: (in class)

 Remarks: We can that the frequencies F_k=F₀+kF_sare

indistinguishable from the frequency F₀after sampling and hence

they are aliases of F₀

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Sampling and Reconstruction

5. Spectrum Replication



x(nT)  x(t)  x(t) (t nT)  x(t)s(t)

s(t)  (t  nT)

 Let

where



n

 s(t) is periodic, thus, its Fourier series are given by



1

S  (t)e^{ j2F nt}dt  (t)dt 

s(t)  S e^{j2 F nt}

s

where



n

_T

T

n

T



1

s

s(t) 

e^{j2 F nt}

Thus,



T

n



1

x(t)e^{j2nf t}

s

x(t)  x(t)s(t) 

which results in



T

n



1

X (F) 

X (F  nF )

x(t)

 Taking the Fourier transform of

yields



s

T

n

 Observation: The spectrum of discrete-time signal is a sum of the

original spectrum of analog signal and its periodic replication at the

interval F_s.

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Sampling and Reconstruction

 F_s/2 ≥ F_max

Fig: Spectrum replication caused by sampling

Fig: Typical badlimited spectrum

 F_s/2 < F_max

Fig: Aliasing caused by overlapping spectral replicas

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Sampling and Reconstruction

6. Sampling Theorem

 For accurate representation of a signal x(t) by its time samples x(nT),

two conditions must be met:

1) The signal x(t) must be band-limited, i.e., its frequency spectrum must

be limited to F_max.

Fig: Typical band-limited spectrum

2) The sampling rate F_smust be chosen at least twice the maximum

F  2F_max

frequency F_max.

s

 F_s=2F_maxis called Nyquist rate; F_s/2 is called Nyquist frequency;

[-F_s/2, F_s/2] is Nyquist interval.

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Sampling and Reconstruction

 The values of F_maxand F_sdepend on the application

Application

Biomedical

Speech

Fmax

Fs

1 KHz

4 KHz

20 KHz

4 MHz

2 KHz

8 KHz

40 KHz

8 MHz

Audio

Video

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Sampling and Reconstruction

7. Ideal analog reconstruction

Fig: Ideal reconstructor as a lowpass filter

 An ideal reconstructor acts as a lowpass filter with cutoff frequency

equal to the Nyquist frequency Fs/2.

T F [F / 2,F / 2]







s

H(F) 

 An ideal reconstructor (lowpass filter)

0

otherwise

Then

X_a(F)  X(F)H(F)  X(F)

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Sampling and Reconstruction

Example 3

 The analog signal x(t)=cos(20πt) is sampled at the sampling

frequency Fs=40 Hz.

a) Plot the spectrum of signal x(t) ?

b) Find the discrete time signal x(n) ?

c) Plot the spectrum of signal x(n) ?

d) The signal x(n) is an input of the ideal reconstructor, find the

reconstructed signal x_a(t) ?

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Sampling and Reconstruction

Example 4

 The analog signal x(t)=cos(100πt) is sampled at the sampling

frequency Fs=40 Hz.

a) Plot the spectrum of signal x(t) ?

b) Find the discrete time signal x(n) ?

c) Plot the spectrum of signal x(n) ?

d) The signal x(n) is an input of the ideal reconstructor, find the

reconstructed signal x_a(t) ?

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Sampling and Reconstruction

 Remarks: x_a(t) contains only the frequency components that lie in the

Nyquist interval (NI) [-F_s/2, F_s/2].

sampling at F_s

 x(t), F₀ NI ------------------> x(n) ----------------------> x_a(t), F_a=F₀

ideal reconstructor

sampling at F_s

ideal reconstructor

 x_k(t), F_k=F₀+kF_s-----------------> x(n) ---------------------> x_a(t), F_a=F₀

 The frequency F_aof reconstructed signal x_a(t) is obtained by adding

to or substracting from F₀(F_k) enough multiples of Fs until it lies

within the Nyquist interval [-F_s/2, F_s/2]. That is

F  F mod(F )

a

s

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Sampling and Reconstruction

Example 5

 The analog signal x(t)=10sin(4πt)+6sin(16πt) is sampled at the rate 20

Hz. Find the reconstructed signal x_a(t) ?

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Sampling and Reconstruction

Example 6

 Let x(t) be the sum of sinusoidal signals

x(t)=4+3cos(πt)+2cos(2πt)+cos(3πt) where t is in milliseconds.

a) Determine the minimum sampling rate that will not cause any

aliasing effects ?

b) To observe aliasing effects, suppose this signal is sampled at half its

Nyquist rate. Determine the signal x_a(t) that would be aliased with

x(t) ? Plot the spectrum of signal x(n) for this sampling rate?

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Sampling and Reconstruction

Example 7

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Sampling and Reconstruction

Bài giảng Digital signal processing - Chapter 1: Sampling and Reconstruction - Nguyễn Thanh Tuấn