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Dr. Jirabhorn Chaiwongsaiดร.จริาพร ไชยวงศ์สาย
Department of Computer Engineering
School of Information and Communication Technology
University of Phayao
Discrete-valued Signals and Sampling Theorem
Voicing Detector
Aj. Jirabhorn Chaiwongsai 2
Example of high level of breath noise produced at the end of speaking, caused by the speaker’s heavy breathing
Source: L. Rabiner, Biing-Hwang Juang, “Fundamentals of speech recognition”, Prentice hall: New Jersey, 1993.
Voicing Detector (Cont.)
Aj. Jirabhorn Chaiwongsai 3
Time domain
Normalization
Zero crossing rate
Energy
Frequency domain
Low pass filter
High pass filter
Band pass filter
Continuous-valued vs Discrete-valued Signals
Aj. Jirabhorn Chaiwongsai 4
Continuous-valued Signals
If a signal takes on all possible values on a finite or an infinite range, it is said to be a continuous-valued signals
Discrete-valued Signals
If a signal takes on values from a finite set of finite set of possible values, it is said to be a discrete-valued signals
Basic part of an analog-to-digital converter
Aj. Jirabhorn Chaiwongsai 5
Sampler Quantizer Coder
Analog Discrete-time Quantized Digitalsignal signal signal signal
)(txa )(nx )(ntxq 01011….
Continuous-valued vs Discrete-valued Signals
Aj. Jirabhorn Chaiwongsai 6
Continuous-valued vs Discrete-valued Signals
Aj. Jirabhorn Chaiwongsai 7
Speech waveform
Aj. Jirabhorn Chaiwongsai 8
Figure 2.1 Plots of a speech waveform: (a) plotted as a continuous-time signal (with MATLAB plot( ) function);(b) plotted as a sampled signal (with MATLAB stem( ) function).
Aj. Jirabhorn Chaiwongsai 9
Sampling rate
Aj. Jirabhorn Chaiwongsai 10
Continuous-time sinusoidal
Discrete-time sinusoidal
)2cos()( 0 tFAtxa
)2cos()( 0 nfAnx
Periodic sampling of an analog signal
Aj. Jirabhorn Chaiwongsai 11
Sampling rate
Aj. Jirabhorn Chaiwongsai 12
Sampling theorem: If the highest frequency contained in an analog signal is
and the signal is sampled at a rate
The sampling rate is called Nyquist rate
BF max
BFFs 22 max
max22 FBFN
)(txa
Example 1
Aj. Jirabhorn Chaiwongsai 13
Consider the analog signal
What is the Nyquist rate for this signal?
Example 2
Aj. Jirabhorn Chaiwongsai 14
Consider the analog signal
a) Determine the minimum sampling rate required to avoid aliasing
b) Suppose that the signal is sampled at the rate
Hz. What is the discrete-time signal
obtained after sampling?
ttxa 100cos3)(
200sF
Quantization
Aj. Jirabhorn Chaiwongsai 15
If N bits are used to represent the value of x(n), then there are distinct value that x(n) can assume
q = ∆ = quantization level
= maximum value of x(n)
= minimum value of x(n)
Quantization error
N2
Mx
mx
2)(
2
qne
Aj. Jirabhorn Chaiwongsai 16
Sampling and quantization of a sinusoidal signal
Aj. Jirabhorn Chaiwongsai 17
Speech signals
Aj. Jirabhorn Chaiwongsai 18
Illustration of quantization
Aj. Jirabhorn Chaiwongsai 19
Example 3
Aj. Jirabhorn Chaiwongsai 20
n x(n) Discrete-time signal
xq(n) Rounding
eq(n) = xq(n)-x(n)Rounding
0 1 1.0 0.0
1 0.9 0.9 0.0
2 0.81 0.8 -0.01
3 0.729 0.7 -0.029
4 0.6561 0.7 0.0439
5 0.59049 0.6 0.00951
6 0.531441 0.5 -0.031441
7 0.4782969 0.5 0.0217031
8 0.43046721 0.4 -0.03046721
9 0.387420489 0.4 0.012579511
Example 3
Aj. Jirabhorn Chaiwongsai 21
Find the number of N bits quantization of the input x(n) where q = 0.1