通信原理(英文版)4PPT课件
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Most popular A/D conversion method: Pulse Code Modulation (PCM)
1
4.2 Sampling of analog signal
4.2.1 Sampling of low-pass analog signal
Usually sampling at equal time interval T Theoretically,
Analog signal
s(t)
2
Proof of sampling theorem Let: s(t) － signal with highest frequency less than fH
T(t) － periodical unit impulse with repetition period T and repetition frequency fs = 1/T, then the sampled signal is:
The above equation shows that spectrum Sk(f) of the sampled signal is the superposition of infinite spectra S(f) with frequency interval fs, since S(f - nfs) is the result of displacement nfs of the signal frequency spectrum S(f) on the
sampling process ＝ periodical unit impulse analog signal Practically,
sampling process ＝ periodical narrow pulse analog signal Sampling theorem: If the highest frequency of a continuous
(f ) is the frequency
spectrum of periodical unit impulse, it can be found as
(f)T1n (f nsf)
3
Substituting (f)T1n(f nfs) into Sk(f)S(f) (f)
obtain S k(f)T 1 S (f)n (f n fs) T 1 S (f n fs)
Chapter 4 Digitiation of analog signal
4.1 Introduction
Two categories of information sources: analog signal, digital signal
Three steps of A/D conversion: sampling, quantization , coding
analog signal s(t) is less than fH , and if it is sampled by
periodic impulses with interval time T 1/2fH , then s(t) can be completely decided by these samples.
with cut-off frequency fH .
➢ Time domain: When the ideal low-pass filter is excited by the sampled pulse sequence, the output of the filter is the sum of the impulse responses, as shown in the figure. The sum of these impulse responses composes the original signal.
is not superposed with each other, as shown in the figure.
Thus frequency spectrum S(f) of signal s(t) can be separated from Sk(f), and s(t) can be easily obtained from S(f), i.e. the
5
Method of restoration of original signal from sampled signal:
➢ Frequency domain: When fs 2fH , the original signal can be
separated from the sampled signal by an ideal low-pass filer
sk(t)s(t)T(t) s(k)T
Let the Fourier transform of sk(t) is Sk(f), then Sk(f)S(f) (f)
where,
Sk(f) － spectrum of sk(t) S(f) － spectrum of s(t) ( f ) － spectrum of T(t)
frequency axis.
The highest frequency of signal s(t) has been assumed less than fH, therefore if frequency interval fs 2fH, then each displaced spectrum S(f) of the original signal contained in Sk(f)
original signal can be restored from the sampled signal.
4Βιβλιοθήκη Baidu
Here, the condition of restoration of the origial signal is
fs 2 fH
2fH is called Nyquist sampling rate. The corresponding smallest sampling time interval is called Nyquist sampling interval.
1
4.2 Sampling of analog signal
4.2.1 Sampling of low-pass analog signal
Usually sampling at equal time interval T Theoretically,
Analog signal
s(t)
2
Proof of sampling theorem Let: s(t) － signal with highest frequency less than fH
T(t) － periodical unit impulse with repetition period T and repetition frequency fs = 1/T, then the sampled signal is:
The above equation shows that spectrum Sk(f) of the sampled signal is the superposition of infinite spectra S(f) with frequency interval fs, since S(f - nfs) is the result of displacement nfs of the signal frequency spectrum S(f) on the
sampling process ＝ periodical unit impulse analog signal Practically,
sampling process ＝ periodical narrow pulse analog signal Sampling theorem: If the highest frequency of a continuous
(f ) is the frequency
spectrum of periodical unit impulse, it can be found as
(f)T1n (f nsf)
3
Substituting (f)T1n(f nfs) into Sk(f)S(f) (f)
obtain S k(f)T 1 S (f)n (f n fs) T 1 S (f n fs)
Chapter 4 Digitiation of analog signal
4.1 Introduction
Two categories of information sources: analog signal, digital signal
Three steps of A/D conversion: sampling, quantization , coding
analog signal s(t) is less than fH , and if it is sampled by
periodic impulses with interval time T 1/2fH , then s(t) can be completely decided by these samples.
with cut-off frequency fH .
➢ Time domain: When the ideal low-pass filter is excited by the sampled pulse sequence, the output of the filter is the sum of the impulse responses, as shown in the figure. The sum of these impulse responses composes the original signal.
is not superposed with each other, as shown in the figure.
Thus frequency spectrum S(f) of signal s(t) can be separated from Sk(f), and s(t) can be easily obtained from S(f), i.e. the
5
Method of restoration of original signal from sampled signal:
➢ Frequency domain: When fs 2fH , the original signal can be
separated from the sampled signal by an ideal low-pass filer
sk(t)s(t)T(t) s(k)T
Let the Fourier transform of sk(t) is Sk(f), then Sk(f)S(f) (f)
where,
Sk(f) － spectrum of sk(t) S(f) － spectrum of s(t) ( f ) － spectrum of T(t)
frequency axis.
The highest frequency of signal s(t) has been assumed less than fH, therefore if frequency interval fs 2fH, then each displaced spectrum S(f) of the original signal contained in Sk(f)
original signal can be restored from the sampled signal.
4Βιβλιοθήκη Baidu
Here, the condition of restoration of the origial signal is
fs 2 fH
2fH is called Nyquist sampling rate. The corresponding smallest sampling time interval is called Nyquist sampling interval.