Introduction to Digital Signal Processing (Discrete...
Transcript of Introduction to Digital Signal Processing (Discrete...
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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
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• 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.
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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
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• 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)
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Discrete-time signal • sequences can often arise from
periodic sampling of an analog signal.
∞<<∞= n-nTxx a ],[
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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.).
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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]
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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
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• 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)
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Course Information
• Teaching assistant: – Yin-Tzu Lin 林映孜
[email protected] • Course webpage: (to determine)
– www.cmlab.csie.ntu.edu.tw/~dsp/dsp2011 • Grades
– Homework x 2 (30%) – Test x 2 (40%) – Term project (30%)
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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.
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• However, sine or cosine wave use different operations to represent signals under amplitude and phase changes.
Amplitude: by multiplication Phase: by addition
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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 =+
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• 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)
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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
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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
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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