Discrete-valued Signals

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

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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.)

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Time domain

Normalization

Zero crossing rate

Energy

Frequency domain

Low pass filter

High pass filter

Band pass filter

Continuous-valued vs Discrete-valued Signals

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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

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Sampler Quantizer Coder

Analog Discrete-time Quantized Digitalsignal signal signal signal

)(txa )(nx )(ntxq 01011….

Continuous-valued vs Discrete-valued Signals

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Continuous-valued vs Discrete-valued Signals

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Speech waveform

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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).

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Sampling rate

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Continuous-time sinusoidal

Discrete-time sinusoidal

)2cos()( 0 tFAtxa

)2cos()( 0 nfAnx

Periodic sampling of an analog signal

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Sampling rate

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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

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Consider the analog signal

What is the Nyquist rate for this signal?

Example 2

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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

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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

qq

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Sampling and quantization of a sinusoidal signal

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Speech signals

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Illustration of quantization

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Example 3

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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

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Find the number of N bits quantization of the input x(n) where q = 0.1