Introduction to Digital Signal Processing

(Discrete-time Signal Processing)

Prof. Chu-Song Chen Research Center for Info. Tech. Innovation, Academia

Sinica, Taiwan Dept. CSIE & GINM

National Taiwan University

Fall 2011

• In our technical society we often measure a continuously varying (analog) quantity. eg. Blood pressure, earthquake displacement, population of a city, waves falling on a beach, and the prob. of death.

• All these measurement varying with time; we regard them as functions of time: x(t) in mathematical notation.

Signals

→ flow of information → measured quantity that varies with time (or

position) → electrical signal received from a transducer (microphone, thermometer, accelerometer, antenna, etc.) → electrical signal that controls a process

• For technical reasons, instead of the signal x(t), we usually record equally spaced samples xn of the function x(t). (discrete-time) – The sampling theorem gives the conditions on the signal

that justify this sampling process. – i.e., discrete-time signal is a sequence of numbers

• Moreover, when the samples are taken they are not recorded with infinite precision but are rounded off (sometimes chopped off) to comparatively few digits.

• This procedure is often called quantizing the samples. (digital)

Discrete-time signal • sequences can often arise from

periodic sampling of an analog signal.

∞<<∞= n-nTxx a ],[

Signal Source – where it comes

• Continuous-time signals: voltage, current, temperature, speed, . . .

• Discrete-time signals: daily minimum/maximum temperature, lap intervals in races, sampled continuous signals, . . . – Electronics can only deal easily with time-

dependent signals; therefore spatial signals, such as images, are typically first converted into a time signal with a scanning process (TV, fax, etc.).

The concept of System

• Signal Processing System: map an input signal to an output signal – Continuous-time systems

• Systems for which both input and output are continuous-time signals

– Digital system • Both input and output are digital signals

x[n] T{⋅} y[n]

Course Outline • Basic topics

→ Z-transform → Discrete-time Fourier transform (DTFT) → Sample of continuous-time signals → Discrete-time linear systems & its transform domain analysis → Structure for discrete-time systems → Digital filter → Discrete Fourier transform (DFT) → Fast computation of discrete Fourier transform → Fourier analysis of signals using DFT → Random signals and systems

• Miscellaneous topics → Gaussian process; Smoothing splines; Wavelets → Bilateral filtering; Total variation → Particle filtering → Machine learning for signal processing

• Reference Textbooks – Alan V. Oppenheim and Ronald W. Schafer, Discrete-

Time Signal Processing, Prentice-Hall. – Sanjit K. Mitra, Digital Signal Processing: A Computer-

based Approach, McGraw Hill • Main Journals

– IEEE Trans. Signal Processing – IEEE Signal Processing Magazine

• Main Conferences – IEEE International Conference on ASSP (ICASSP)

Course Information

• Teaching assistant: – Yin-Tzu Lin 林映孜

known@cmlab.csie.ntu.edu.tw • Course webpage: (to determine)

– www.cmlab.csie.ntu.edu.tw/~dsp/dsp2011 • Grades

– Homework x 2 (30%) – Test x 2 (40%) – Term project (30%)

Introduction to complex exponentials

• Signal processing is originated form the processing of “frequency.”

• We hope to decompose the signals by extracting its components with respect to different frequencies. – Important to the field of broadcasting, wireless

communication, music analysis, etc. • Basically, frequency stems from the periodic

sinusoidal function (sine or cosine waves). – Eg., x(t)=sin(w0t); frequency w0; period 2π/w0. – sin(w0t+φ); frequency w0; phase φ.

• However, sine or cosine wave use different operations to represent signals under amplitude and phase changes.

Amplitude: by multiplication Phase: by addition

Complex-number signals • In signal processing, it is quite often to use complex

function to represent a signal: • complex exponentials:

where

• w0 is called the frequency of the complex exponential and φ is called the phase.

• To represent discrete-time signals, we sample uniformly the function into n points within the 2π period,

( )φ+twje 0

( ) )sin()cos( 000 φφφ +++=+ twjtwe twj

( ) nwe mwj /2, 00 πφ =+

• By using complex exponential, a further way is to use the same operation (multiplication) to represent both amplitude and phase changes.

Amplitude and Phase: by multiplication

( ) tjwjtwj eAeAe 00 )( φφ =+

complex number multiplication can represent both scaling (amplitude variation) and rotation (phase shift)

Further advantage of using complex exponential

geometric series is used quite often to simplify expressions in DSP.

if the magnitude of x is less than one, then

xxxxxx

NN

n

Nn

−−

=++++=∑−

=

−

111

1

0

12

1 ,1

10

<−

=∑∞

=

xx

xn

n

Note that rigonometric functions, especially sine and cosine functions, appear in different combinations in all kinds of harmonic analysis: Fourier series, Fourier transforms, etc. Advantages of complex exponential The identities that give sine and cosine functions in terms of exponentials are important – because they allow us to find sums of sines and cosines using the geometric series. Eg. we know ie. a sum of equally spaced samples of any sine or cosine function within 2π is zero, provided the sum is over a cycle (or a number of cycles), of the function.

∑−

=

=

1

002sin

N

n Nnπ ∑

−

=

=

1

002cos

N

n Nnπ

It can be more easily verified by the geometrical sequence of complex exponential

01

11

02

22

=−

−=∑

−

=

N

n Nnj

jN

nj

e

ee π

ππ