Discrete-time Signals & Systems

S Wongsa

11

S WongsaDept. of Control Systems and Instrumentation Engineering,

KMUTT

JAN, 2011

Overview

� Signals & Systems

� Continuous & Discrete Time

� Sampling

� Sampling in Frequency Domain

� Sampling Theorem

22

� Sampling Theorem

� Aliasing & Anti-Aliasing Filter

Lecture plan

Lecture Date Topic

1 4 & 5 Jan 11 Discrete-time signals and systems; Sampling of continuous-time signals

2 11 & 12 Jan 11 Discrete-Time Fourier Transform (DTFT) & Discrete-Fourier Transform (DFT)

3 18 & 19 Jan 11 Fast Fourier Transform (FFT) & Applications (Lab I)

4 25 & 26 Jan 11 z-Transform

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5 1 & 2 Feb 11 Transform-domain analysis and LTI systems

6 8 & 9 Feb 11 Discrete-time system analysis Lab (II)

7 15 & 16 Feb 11 Tutorial

Course website: http://webstaff.kmutt.ac.th/~sarawan.won/INC212/

Grading

1.) Graded homework worth 10%2.) Laboratory assignments worth 10%

3.) Final MATLAB exam worth 5%4.) Final exam worth 25%

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Recommended Textbooks

1. Fundamentals of Signals and Systems Using MATLAB, Edward W. Kamenand Bonnie S. Heck, Prentice Hall International Inc.

2. Discrete-time signal processing, A.V. Oppenheim, R.W. Schafer, and J. R. Buck, 2nd edition, Prentice Hall, 1999.

3. Signals and Systems, Alan v. Oppenheim et al., 2nd Edition, Prentice Hall.

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4. Signals and Systems, Simon Haykin & Barry Van Veen, 2nd edition, Wiley, 2003.

Signals & Systems

A signal is a varying phenomenon that can be measured.

A system responses to particular signals by producing other signals.

66Source: 6.003 Signals & Systems, MIT, Fall 2009.

Signals & Systems

An image is also a signal!

77Source: Yao Wang, Introduction, Review of Signals & Systems, Image Quality Metrics, Polytechnic University, Brooklyn, NY

Discrete-time processing of continuous-time signals

Sampling Reconstruction

e.g. DSP,Controller etc.

88Source: 6.003 Signals & Systems, MIT, Fall 2009.

• Most of the signals in the physical world are CT signals, e.g. voltage & current, pressure, temperature, velocity, etc.

• But digital computations are done in discrete time.

Discrete-time processing of continuous-time signals

99Source: Prof. Mark Fowler, EECE 301 Signals & Systems, Binghamton University.

Discrete-time processing of continuous-time signals

Sampling Reconstruction

e.g. DSP,Controller etc.

1010

Controller etc.

DSP for Detection of Weld Defects

Original image After DSP

Defects ?

1111Source: W.Yuttiwat et al., Visual Inspection of Weld Defects by Radiography Image Processing , IE Network 2010.

Incomplete penetration

Porosity

DSP: Biomedical Imaging

X-Ray CT MRI – Magnetic

Resonance Imaging

1212Source: http://old.lf3.cuni.cz/biofyzika/doc/eq/ImagingMethods.pp

http://www.diffen.com/difference/CT_Scan_vs_MRI

• make use of radiation to get an internal

view of the body.

• uses magnetic fields in conjunction

with radio waves to give high detail in

the soft tissues.

• use a series of X-ray beams to create cross-sectional images.

• be blocked by some form of dense

tissue, therefore the image quality

when looking at soft tissues will be poor.

• DSP is used to generate a

3D image of the internals of an

object from a large series of 2D X-ray images taken around

a single axis of rotation

• No biological hazards have been

reported with the use of the MRI.

• can pose the risk of irradiation.

Audio Signal Processing

• Music

• Speech Generation

e.g. Text-to-Speech Synthesis, Voice conversion

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• Speech Recognition

Source: http://www.busim.ee.boun.edu.tr/~speech/projects/Voice_Conversion.htmhttp://www2.research.att.com/~ttsweb/tts/demo.php

Discrete-time processing of continuous-time signals

Sampling Reconstruction

1414Source: 6.003 Signals & Systems, MIT, Fall 2009.

Sampling

Sampling is the process of getting a discrete signal from a continuous one.

It enables the processing of signal by digital computer.

T

)(tx )(txs

1515

• Discrete-time signal

K,2,1,0 ],[)()( ±±=== nnxnTxtxs

T

where T is a sampling time.

Sampling

We would like to sample in a way that preserves information, whichmay not seem possible because information between samples is lost.

1616Source: 6.003 Signals & Systems, MIT, Fall 2009.

How can we minimise the distortion of reconstructed signal?

Sampling

)(tx

)(txs

)(tTδ

X

1717

)()()( ttxtxTsδ=

where

∑∞

−∞=

−=n

TnTtt )()( δδ

Sampling in frequency domain

The Fourier transform of : )(txs

∑∞

−∞=

−=k

sskX

TX )(

1)( ωωω

where

Ts

πω

2= is the sampling frequency in rad/sec.

1818

T

Determine under what conditions we get:Reconstructed CT signal = Original CT signal

Goal:

Sampling in frequency domain

• If x(t) has bandwidth B and if Bs

2>ω

∑∞

−∞=

−=k

sskX

TX )(

1)( ωωω

x(t) is a bandlimited signal.

T

1919

The high frequency copies can be removed with a low-pass and then multiplying by T to undo the amplitude scaling.

T

Sampling theorem

A bandlimited signal with bandwidth B can be reconstructed completely and

exactly from its samples as long as they are taken at rate Bs

2>ω

• is called the Nyquist sampling frequency / Nyquist rate.Bs

2=ω

2020

NB: Sampling at Nyquist rate is only possible if an IDEAL lowpass filter is used. In practice we generally need to choose a sampling rate above the Nyquist rate.

What if the samples are not taken fast enough?

B-B

)(ωs

X

Aliasing Aliasing

2121

The high frequency components of x(t) will be transposed to low-frequency components, leading to a phenomenon called aliasing.

What if the signal is not bandlimited?

2222

• For non-bandlimited signal aliasing always happens regardless of value.sω

Aliasing

• What are the consequences of aliasing?

- it makes two continuous sinusoids of different frequencies indistinguishable when sampled.

2

3

2323

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

Time (sec)

Am

plit

ude

Aliasing: a 52 Hz sinusoid sampled at 50 Hz.

Aliasing

• What are the consequences of aliasing?

- a distorted version of the original signal x(t).

• original music sampled at 44.1kHz (CD-quality)

Example:

• The at 4kHz downsampled version.

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• The at 4kHz downsampled version.

Anti-Aliasing FilterTo avoid aliasing, in practice we use a CT lowpass filter before the ADC to restrict the bandwidth of a signal to approximately satisfy the sampling theorem.

Fs = 44.1 kHz

2525Source: Prof. Mark Fowler, EECE 301 Signals & Systems, Binghamton University.

Suggested Readings

• Steven W. Smith, Chapter 3: ADC and DAC, The Scientist and Engineer's Guide to Digital

Signal Processing

2626

Review Questions

1. If we used x(t) below and sampled it at 20 kHz, how many samples would we have after 60 ms?

)5/6602cos()65102cos(2)4/4042cos(3)( πππππ −+++= tttx

3. A periodic signal with a period of 0.1 ms is sampled at 44 kHz. To what

2. x(t) = 2 cos(2π700t − 5π/2) + 3 cos(2π450t) + cos(2π630t + 2π/5)What is the minimum sampling rate for this signal?

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3. A periodic signal with a period of 0.1 ms is sampled at 44 kHz. To what frequency does the eighth harmonic alias?

Summary

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