电子科大数字信号处理课件

Chapter4 Digital Processing of Continuous-Time SignalsSampling of Continuous-Time SignalsRecovery of the Analog Signal

Implication of the Sampling ProcessSampling of Bandpass Signals1

Analog Lowpass Filter Design

电子科大数字信号处理课件

4.1 Introduction

The Basic Framework of a DSP Systemx[n] AD DSP y[n] DA Analog filterya (t )

xa (t )

Pre-filter S/H

1.Pre-filter:get rid of the frequency components we don’t interested in.—Anti-aliasing filter. 2.ADC:transform analog signal into digital signal. 3.DSP:process the digital signal. 4.DA:transform digital signal into analog ones. 5.Analog filter:filter the high frequency that is needless.—Reconstruction filter.

电子科大数字信号处理课件

Introduction

A simplified representation of DSP system:Ideal Samplerx[n]

xa (t )

Discrete Time Processor

y[n]

Ideal Interpolator

ya (t )

We will use this figure to represent the working principle of this system.

电子科大数字信号处理课件

4.2 Sampling of Continuous-Time Signalsg a (t )

g p (t )

g a (t )

g p (t )

p(t )

(a)g a (t )0

Ideal sampling modelst

(b)

g a (t )0

t

p(t)t 0 T t

p(t) T (t ) 0 T

t

g p (t )

g p (t )t 0

4

0

电子科大数字信号处理课件

4.2.1 Effect of Sampling in the Frequency DomainWhen τ «T, p(t) ~δT(t),so firstly to discuss ideal sampling: (t)= (t nT ) Tn

The output of ideal sampling is: (1)In time domain:g p (t ) ga (t ) T (t ) 5n

g

a

(nT ) (t nT )

n

g

a

(t ) (t nT )

电子科大数字信号处理课件

Effect of Sampling in the Frequency Domain(2)In frequency domain: Ga ( j ) CTFT [ g a (t )] If given: T ( j ) CTFT [ T (t )] G p ( j ) CTFT [ g p (t )]

We can get:1 GP ( j ) Ga ( j ) T ( j ) 2

2 T ( j ) s ( j jk s ), s T k

电子科大数字信号处理课件

Effect of Sampling in the Frequency Domain1 1 2 G p ( j ) Ga ( j jk s ) Ga ( j jk ) T k T k TGa ( j )

0

… s0

s

The frequency procedure of sampling…

s

2 s

G p ( j )

… 7 s 0 s 2 s

…

电子科大数字信号处理课件

Sampling Theorem-(Nyquist Theorem)

Assume ga(t) is a band-limited signal with a CTFT Ga(j ) as shown below:

The spectrum P(j ) of p(t) having a sampling period T=2 / T is indicated below:

电子科大数字信号处理课件

Sampling Theorem-(Nyquist Theorem)

Two possible spectra of Gp(j ) are shown below:

电子科大数字信号处理课件

Sampling Theorem-(Nyquist Theorem)From above discussion, we can see: (1)If ga(t) is bandlimited with

(2)If

s 2 m ( s

Ga ( j ) 0 for m 2 T )

Please look at P175 Fig4.4.(a)(b)(c) We call the frequency Ωs/2 which satisfies condition(2) is Nyquist frequency or folding frequency, and (2) is called Nyquist condition.10

电子科大数字信号处理课件

Sampling Theorem-(Nyquist Theorem)So let’s see sampling theorem: If want to reconstruct the signal after sampling, the sampling frequency must be greater than two times of the highest frequency o

f the signal.that is : fs>2fh. Several concepts:(P175 Figure4.4) Over-sampling ,Under-sampling , criticalsampling11

电子科大数字信号处理课件

Sampling Theorem-(Nyquist Theorem)

Over-sampling - The sampling frequency is higher than the Nyquist rate. Under-sampling - The sampling frequency is lower than the Nyquist rate. Critical sampling - The sampling frequency is equal to the Nyquist rate. Note: A pure sinusoid may not be recoverable from its critically sampled version.

电子科大数字信号处理课件

Sampling Theorem-(Nyquist Theorem)

In digital telephony, a 3.4 kHz signal bandwidth is acceptable for telephone conversation. Here, a sampling rate of 8 kHz, which is greater than twice the signal bandwidth, is used.

电子科大数字信号处理课件

Sampling Theorem-(Nyquist Theorem)

In high-quality analog music signal processing, a bandwidth of 20 kHz has been determined to preserve the fidelity. Hence, in compact disc (CD) music systems, a sampling rate of 44.1 kHz, which is slightly higher than twice the signal bandwidth, is used.

电子科大数字信号处理课件

Example

Consider the three continuous-time sinusoidal signals: g1(t ) cos( 6 t ) g 2 (t ) cos(14 t ) g3 (t ) cos( 26 t ) Their corresponding CTFTs are: G1 ( j ) [ ( 6 ) ( 6 )]G2 ( j ) [ ( 14 ) ( 14 )]G3 ( j ) [ ( 26 ) ( 26 )]

电子科大数字信号处理课件

ExampleThese three transforms are plotted below: