디지털신호처리 digital signal processing

디지털신호처리DIGITAL SIGNAL PROCESSING전남대학교 공과대학 전자컴퓨터공학부 김진영2010.3.

DSP Introduction

왜 DSP 인가 ? : 통신의 역사 1

1790 semaphore lines

490 BC heliograph19C signal lamp

1150 homing pigeon


우리나라의 통신정낭 : 3bits

봉화


1895 년 마르코니(Marconi) 는 최초

의 무선 시스템을 시작하였다 . 마르

코니의 초기spark-gap 송신기

는 매우 낮은 주파 수에서부터 단파대

이상 까지의 넓은 스펙트럼을 점유하

였으며 이들 시스템 은 수동으로 시간

도메인에서 모르스 부호를 사람들이 송

· 수신함으로써 동작하였다 .

왜 DSP 인가 ?

All communications circuits contain some noise. This is true whether the signals are analog or digital, and regardless of the type of information conveyed. Noise is the eternal bane of communications engineers, who are always striving to find new ways to improve the signal-to-noise ratio in communications systems. Traditional methods of optimizing S/N ratio include increasing the transmitted signal power and increasing the receiver sensitivity. (In wireless systems, specialized antenna systems can also help.) Digital signal processing dramatically improves the sensitivity of a receiving unit. The effect is most noticeable when noise competes with a desired signal. A good DSP circuit can sometimes seem like an electronic miracle worker. But there are limits to what it can do. If the noise is so strong that all traces of the signal are obliterated, a DSP circuit cannot find any order in the chaos, and no signal will be received.

Battle against Noise

Contents

DSP definition and application Frequency and sinusoids Discrete time signals Basic DSP operations Discrete time systems DSP system

Digital Signal

What is Signal? Any physical quantities that varies with time,

space or any other independent variable Carriers of information

Classification of signals Scalar vs. vector Continuous-time vs. discrete-time Continuous-valued vs. discrete-valued Deterministic vs. random

Digital Signal=discrete-time+discrete-valued

DSP 정의 및 응용

System and Signal processing System

a Physical device that performs an operation on a signal : Extracting or enhancing the useful information from a mix of conflicting information

Signal processing Any operation on signal : software and

hardware Digital Signal Processing : any operation on

digital signal


Why DSP

Guaranteed accuracy: better control of accuracy requirements

Perfect reproducibility Stable processing capability: no drift in

performance with temperature or age Greater flexibility Superior performance Cheaper Portability: using software running


Disadvantage of DSPDSP 정의 및

응용

Speed Bandwidths in the 100MHz range is still

processed only by analogue device Design time Quantization error Finite wordlength effect

DSP Category

Digital Signal

Analysis Digital filter

∙Spectrum analysis∙Speech Recognition∙Speaker verification∙Target detection

∙Removal of unwanted background noise∙Removal of interference∙Separation of frequency bands∙Shaping of the signal spectrum

ApplicationDSP 정의 및

응용

Image processing: pattern recognition, robotic vision, image enhancement, facsimile, satellite weather map, animation

Instrumentation/control: spectrum analysis, position and rate control, noise reduction, data compression

Speech/Audio: speech recognition, speech synthesis, digital audio, equalization

Military: secure communication, radar & sonar processing, missile guidance

Telecommunications: echo cancellation, adaptive equalization, codec, spread spectrum, video conference, data communication

Biomedical : patient monitoring, scanners, EEG brain mapers, ECG analysis, X-ray storage/enhancement

Etc

Frequency?주파수 개념 및

삼각함수들

Definition 1. Physics: The rate at which a repeating event

occurs, such as the full cycle of a wave. Frequencies are usually measured in hertz. Compare amplitude. See also period.

2. Mathematics: The ratio of the number of occurrences of some event to the number of opportunities for its occurrence.

Why sinusoids?

The most basic signals in the theory of signals and systems

Sinusoidal function is an eigen-function of the linear system

주파수 개념 및 삼각함수들

Continuous/discrete-time sinusoidal signal

Continuous-time

Discrete-time


Continuous/discrete-time sinusoidal signal


Phasor 1

Phasor vector: a representation of a sine wave whose amplitude(A), phase(θ) and frequency(ω)

Rotating phasor interpretation

Phasor addition


))(( )(~)(

)(~

0)(

)(

0

00

ttAetxAeXXe

eAeAetx

tj

jtj

tjjtj

)180/79.141)10(2cos(532.1)()()()180/200)10(2cos(9.1)(

)180/70)10(2cos(7.1)(

213

2

1

ttxtxtxttxttx

Phasor 2

Phasor arithmetic Scalar multiplication

Differentiation and integration addition


Phasor 3

http://en.wikipedia.org/wiki/Phasor_(electronics)


Types of Sequence

Unit sample sequence Unit step function Real-valued exponential sequence

(Geometric series) Complex-valued exponential sequence Sinusoidal sequence Random sequence Periodic sequence

이산 - 시간 신호

Some useful sequence

Unit sample synthesis Even and odd synthesis

이산 - 시간 신호

( ) ( ) ( )x n x k n k

( ) ( ), ( ) ( )

( ) ( ) ( )

1( ) ( ) ( )

21

( ) ( ) ( )2

e e o o

e o

e

o

x n x n x n x n

x n x n x n

x n x n x n

x n x n x n

DSP Operation 1

Signal addition Signal multiplication Scaling Time shifting and advancing Sample summation Sample product

Signal energy and power (signal measure)

주요 DSP 조작

1 2( ) ( )x n x n

1 2( ) ( )x n x n

( )x n( ) ( )

( ) ( )

y n x n k

y n x n k

2

1

( )n

n n

x n

2

1

( )n

n n

x n

* 2 2

0

1( ) ( ) | ( ) | , | ( ) |

N

x xn

E x n x n x n P x nN

DSP Operation 2

Convolution

Correlation

Digital filter

Discrete transform: discrete Fourier transform

Modulation: digital modulation

주요 DSP 조작

( ) ( ) ( ) ( ) ( )k

y n h n x n h k x n k

( ) ( ) ( ) ( ) ( )k

y n h n x n h k x k n

0

( ) ( ) ( )n

k

y n h k x n k

Properties of discrete time system1 Static v.s dynamic systems

Static: memoryless

Dynamic: memory

Time-invariant vs. time-variant systems (Def) Time-invariant(shift invariant) iff

x(n)→y(n) implies that x(n-k) →y(n-k) for every input signal x(n) and every time shift

(test) y(n,k)=H[x(n-k)] if y(n,k)=y(n-k) for all possible k: time invariant

Discrete time system

3( ) ( ), ( ) ( ) ( )

( ) [ ( ), ]

y n a x y n nx n bx x

y n F x n n

0

( ) ( ) 3 ( 1); finite memory

( ) ( ); infinite memoryk

y n x n x n

y n x n k

Properties of discrete time system 2

(Ex) time-multiplier

Linear vs. nonlinear systems (Def) A system H is linear iff H[a1x1 (n)+a2x2 (n)]=a1H[x1 (n)]+a2H[x2 (n)]

( ) [ ( )] ( )

1) ( , ) [ ( )] ( )

2) ( ) ( ) ( ) ( ) ( )

1) 2) time variant

y n h x n nx n

y n k H x n k nx n k

y n k n k x n k nx n k kx n k


Properties of discrete time system 3 Causal vs. noncausal systems

(Def) causal if the output of the system at any time n depends only on present and post inputs; y(n)=F[x(n),x(n-1),x(n-2),….]

(ex) y(n)=x(-n): noncausal Stable vs. unstable system

(Def) Bounded input-bound output(BIBO) iff every bounded input produces a bounded output;


| ( ) | | ( ) |x yx n M y n M

Properties of discrete time system 4 Invertible linear system

(TH) L:X→Y be an invertible linear transformation of X onto Y, where X, Y are linear spaces, then L-1 is linear

(proof)


Linear System 1

General linear system

Linear time-invariant (LTI) system: an input-output pair, x(n) and y(n), is invariant to shift in time n.


( ) [ ( )] [ ( ) ( )] ( ) [ ( )]k k

y n L x n L x k n k x k L n k

[ ( )]: the response of a linear system at time n due to a unit sample

at time k, denoted by ( , ) ( ) ( ) ( , )k

L n k

h n k y n x k h n k

( ) [ ( )] ( ) ( ) ( ) ( ) ( ) ( )n n

y n L x n x k h n k x n h n h k x n k

Linear System 2

Why difference equations:

Difference equations: An LTI system can be described by a linear constant coefficient difference equation of the form.


0 0

0 1

( ) ( )

( ) ( ) ( )

N M

k mk m

M N

m km k

a y n k b x n m

y n b x n m a y n k

0 0

( ) ( ) ( ) ( )( ) lim lim

( ) (( 1) )

t t

d x t x t t x t t x tx t

dt t tx nT x n T

T

0 0

( ) ( )k kN M

k kk kk k

d da y t b x tdt dt

Block diagram of DSP system A/D conversion DSP DA conversion

Digital signal processing system

Block diagram of a simplified, generalized real-time digital signal

processing system

Analog-to-digital conversion process 1 A/D process

The (band limited) signal is first sampled in time (t=nT, x(nT)→x(n))

The amplitude of each signal sample is quantized into one of 2B levels

The discrete amplitude levels are represented or encoded into distinct binary words each of length B bits


Sampling 1

Sampling Sampling theorem: If the highest frequency

contained in an analog signal xa(t) is Fmax=B and the signal is sampled at a rate Fs≥2fmax=2B, then xa(t) can be exactly recovered from its sample values using the interpolation function g(t)=(sin2πBT)/(2πBT). Thus xa(t) may be expressed as

where xa(n/Fs)=xa(nT)=x(n) are the samples of xa(t).


0

( ) ( / ) ( / )M

a a s sm

x t x n F g t n F

http://en.wikipedia.org/wiki/File:Sinc_function_(normalized).svg

Sampling 2

Xs(f)=XaⓧP(f) where P(f) is periodic signal in frequency domain with period Fs, because

If Fs<Fmax, aliasing occurs. If Fs<Fmax, anti-aliasing filters

are necessary.


( ) ( ), ( ) ( )s s sk k

P f A f F k X f X f F k

0

0

/ 2

/ 2

0

2 /

1 1[ ] ( )

2( ) ( )

Tjk t

T

k

T

P k t e dtT T

P j kT

Sampling 3Digital signal processing

system

Sampling 4Digital signal processing

system

622x756 pixels

205x250 pixels (Moire pattern of bricks)

Quantization and encoding 1 Linear quantization

v(t)=(Vfs/2)sinωt=Asinωt Quantization step=q=Vfs(2B-1)≈Vfs/2B=2A/2B

Quantization error

Signal-to-quantization noise power ratio(SQNR)


2/ 2 / 22 2 2

/ 2 / 2

1( )

12

q q

e q q

qe P e de e de

q

2 2 2

2 2

/ 2 ( 2 / 2) / 2 3 210log 10log 10log

/12 /12 2

6.02 1.76( )

B BA qSQNR

q q

B dB

Quantization and encoding 2

Quantization of a signal for 4bit PCM


Digital-to-analog conversion process: Signal recovery 1 Basic idea: Because there is no ideal low

pass filter, the perfect reconstruction is impossible and the impulse signal is not possible in real world ( 참조 )

Real implementation

Anti-imaging filter: Attenuate the high frequency image spectrum(post-filter)


( ) ( )sinc[ ( )]a s sn

x t x n F t nT

Digital-to-analog conversion process: Signal recovery 2 ( 참조 )

Zero-order hold(ZOH) interpolation


0

( ) ( ), ( 1)

1, 0( )

0, otherwise

a s s

s

x t x n nT t n T

t Th t


Digital-to-analog conversion process: Signal recovery 3

First-order-hold(FOH) interpolation

1

, 0

( )( ) 1 , 2

0, otherwise

ss

ss s

s

tt T

T

t Th t T t T

T

디지털신호처리 digital signal processing

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