# Xử lý tín hiệu số sampling and reconstruction

Exercises 2a • A sound wave has the form where t is in milliseconds. What is the frequency content of this signal? Which parts of it are audible and why?

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Sampling and Reconstruction
Ngô Quốc Cường
Ngô Quốc Cường
ngoquoccuong175@gmail.com
sites.google.com/a/hcmute.edu.vn/ngoquoccuong
Sampling and reconstruction
• Introduction
• Review of analog signal
• Sampling theorem
• Analog reconstruction
2
1.1 Introduction
Digital processing of analog signals proceeds in three
stages:
1. The analog signal is digitized, that is, it is sampled and each
sample quantized to a finite number of bits. This process is
called A/D conversion.
2. The digitized samples are processed by a digital signal
processor.
3. The resulting output samples may be converted back into
analog form by an analog reconstructor (D/A conversion).
3
1.2 Review of analog signal
• An analog signal is described by a function of time, say, x(t).
The Fourier transform X(Ω) of x(t) is the frequency spectrum
of the signal:
• The physical meaning of X(Ω) is brought out by the inverse
Fourier transform, which expresses the arbitrary signal x(t) as
a linear superposition of sinusoids of different frequencies:
4
1.2 Review of analog signal
• The response of a linear system to an input signal x(t):
• The system is characterized completely by the impulse
response function h(t). The output y(t) is obtained in the time
domain by convolution:
5
1.2 Review of analog signal
• In the frequency domain by multiplication:
• where H(Ω) is the frequency response of the system, defined
as the Fourier transform of the impulse response h(t):
6
1.2 Review of analog signal
7
8
CT Fourier Transforms of Periodic Signals
Source: Jacob White
9
Fourier Transform of Cosine
Source: Jacob White
10
Note: (period in t) T
(period in ) 2/T
Impulse Train (Sampling Function)
Source: Jacob White
1.3 Sampling theorem
• The sampling process is illustrated in Fig. 1.3.1, where the
analog signal x(t) is periodically measured every T seconds.
Thus, time is discretized in units of the sampling interval T:
11
1.3 Sampling theorem
• For system design purposes, two questions must be
answered:
1. What is the effect of sampling on the original frequency
spectrum?
2. How should one choose the sampling interval T?
12
1.3 Sampling theorem
• Although the sampling process generates high frequency
components, these components appear in a very regular
fashion, that is, every frequency component of the original
signal is periodically replicated over the entire frequency axis,
with period given by the sampling rate:
• Let x(t) = xc(t), sampling pulse s(t)
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1.3 Sampling theorem
14 Source: Zheng-Hua Tan
1.3 Sampling theorem
15 Source: Zheng-Hua Tan
1.3 Sampling theorem
16 Source: Zheng-Hua Tan
1.3 Sampling theorem
17 Source: Zheng-Hua Tan
1.3 Sampling theorem
18 Source: Zheng-Hua Tan
1.3 Sampling theorem
19 Source: Zheng-Hua Tan
1.3 Sampling theorem
• The sampling theorem provides a quantitative answer to the
question of how to choose the sampling time interval T.
• T must be small enough so that signal variations that occur
between samples are not lost. But how small is small
enough?
• It would be very impractical to choose T too small because
then there would be too many samples to be processed.
20
1.3 Sampling theorem
Hardware limits
• In real-time applications, each input sample must be
acquired, quantized, and processed by the DSP, and the
output sample converted back into analog format. Many of
these operations can be pipelined to reduce the total
processing time.
• In any case, there is a total processing or computation time,
say Tproc seconds, required for each sample.
where:
21
1.4 Aliasing and reconstructor
• The set of frequencies,
are equivalent to each other.
• Among the frequencies in the replicated set, there is a unique
one that lies within the Nyquist interval. It is obtained by
reducing the original f modulo-fs.
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1.4 Aliasing and reconstructor
• Antialiasing prefilters
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1.4 Aliasing and reconstructor
• An ideal analog reconstructor extracts from a sampled signal
all the frequency components that lie within the Nyquist
interval.
24
Exercises 1
• Let x(t) be the sum of sinusoidal signals
25
Exercises 2a
• A sound wave has the form
where t is in milliseconds. What is the frequency content of this
signal? Which parts of it are audible and why?
26
Exercises 2b
27

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