Chapter 1 Introduction - ee.nthu.edu.tw

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Introduction

清大電機系林嘉文 [email protected]

03-5731152

Chapter 1

Signal & Signal Processing

• Signal: quantity that carries information

• Signal Processing is to study how to represent,

convert, interpret, and manipulate a signal and

the information contained in the signal

• DSP: signal processing in the digital domain

2011/2/23 Digital Signal Processing 2

mailto:[email protected]

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Signals & Systems

Signals

“Something” that carries information

Speech, audio, image, video, biomedical signals,

radar signals, seismic signals, etc.

Systems

“Something” that can manipulate, change, record,

or transmit input signals

Examples: CD, VCD/DVD


“Discrete-Time” Signals vs. “Digital”

Signals

Discrete-Time signal

A “sampled” version of a continuous signal

What is the minimum sampling frequency which is enough

to perfectly reconstruct the original continuous signal?

Nyquist rate (Shannon sampling theorem)

Digital Signal

“Sampling” + “Quantization” + “Coding”

Quantization: discrete-time continuous-valued signal ->

discrete-time discrete-valued signal

Coding: use a binary number of finite bits (e.g., 8 bits) to

represent a sampled value


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Examples of Typical Signals

Speech and music signals - Represent air

pressure as a function of time at a point in space

Waveform of the speech signal“ I like digital

signal processing” :


Digital Speech Signals

Voice frequency range: 20Hz ~ 3.4 KHz

Sampling rate: 8 KHz (8000 samples/sec)

Quantization: 8 bits/sample

Bit-rate: 8K samples/sec * 8 bits/sample = 64

Kbps (for uncompressed digital phone)

In current Voice over IP (VOIP) technology,

digital speech signals are usually compressed

(compression ratio: 8~10)


http://upload.wikimedia.org/wikipedia/commons/1/11/Microphone_U87.jpg

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Digital Audio Signals

Frequency range of human hearing system:

20Hz ~ 20 KHz

CD rate is 44,100 samples per second

16-bit samples

Stereo uses 2 channels

Number of bytes for 1 minute is

2 X (16/8) X 60 X 44100 = 10.584 Mbytes

What’s the length that a CD-ROM (680 Mbytes)

can store?

How about MP3?



Dow Jones Industrial Average


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Electrocardiography (ECG) Signal - Represents

the electrical activity of heart


ECG Signal

The ECG trace is a periodic waveform

One period of the waveform shown below

represents one cycle of the blood transfer process

from the heart to the arteries


http://upload.wikimedia.org/wikipedia/en/b/bd/12leadECG.jpg

http://en.wikipedia.org/wiki/File:SinusRhythmLabels.svg

http://en.wikipedia.org/wiki/File:ECG_principle_slow.gif

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Electroencephalogram (EEG) Signals -

Represent the electrical activity caused by the

random firings of billions of neurons in the brain



Seismic Signals - Caused by the movement

of rocks resulting from an earthquake, a volcanic

eruption, or an underground explosion


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Black-and-white image - Represents light

intensity as a function of two spatial coordinates

I (x ,y)



Color Image – Consists of Red, Green, and

Blue (RGB) components


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Surface Search Radar Image


Digital Image

• An one mega-pixel image (1024x1024)

• Quantization: 24 bits/pixel for the RGB full-color

space

• File size of a color image: 1024x1024 pixels x

24 bits/pixel = 24 Mbits = 3 Mbytes (for

uncompressed digital image)

• How many uncompressed images can be

stored in a 2G SD flash-memory card?

• What is the compression ratio of JPEG used in

your digital camera?


http://upload.wikimedia.org/wikipedia/commons/0/0d/Radar_screen.JPG

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Digital Image (Cont.)

• In your image processing course, you were taught how

to do

– Edge detection (high-pass filtering)

– Image blurring or noise reduction (low-pass filtering)

– Object segmentation (spatial coherence classification)

– Image compression (retaining most significant info)

• The above are all about mathematical manipulations

– Could you give mathematical formulations for the

above manipulations?

– Could you characterize the frequency behaviors of

the above operations?

– Could you design an image processing tool to meet a

given spec?


Example of Digital Image Processing

Blurring

Edge Detection Original Image


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Examples of Typical Signals: Video

Video Camera

Video Cable

a single scan line

Voltage (proportional

to brightness)

Time

forehead

waveform of scan line shown

wall wall

active video sync and blanking

Video Monitor

hair hair


Video signals - Consists of a sequence of

images, called frames, and is a function of 3

variables: 2 spatial coordinates and time

Frame 1 Frame 3 Frame 5


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Classifications of Signals (1/4)

Types of signal depend on the nature of the

independent variables and the value of the function

defining the signal

for example, the independent variables can be continuous or

discrete

likewise, the signal can be a continuous or discrete function of

the independent variables

for an 1-D signal, the independent variable is usually labeled as

time

A signal can be either a real-valued function or a

complex-valued function

A signal generated by a single source is called a scalar

signal, whereas a signal generated by multiple sources

is called a vector signal or a multichannel signal



A continuous-time signal is defined at every instant of

time

A discrete-time signal is defined at discrete instants of

time, and hence, it is a sequence of numbers

A continuous-time signal with a continuous amplitude is

usually called an analog signal (e.g., speech)

A discrete-time signal with discrete-valued amplitudes

represented by a finite number of digits is referred to as

a digital signal

A discrete-time signal with continuous-valued amplitudes

is called a sampled-data signal

A continuous-time signal with discrete-value amplitudes

is usually called a quantized boxcar signal


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A signal that can be uniquely determined by a well-

defined process, such as a mathematical expression or

rule, or table look-up, is called a deterministic signal

A signal that is generated in a random fashion and

cannot be predicted ahead of time is called a random

signal


Classification of Signals (4/4)


quantized boxcar

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Why DSP?

Mathematical abstractions lead to

generalization and discovery of new processing

techniques

Computer implementations are flexible

Applications provide a physical context


Advantages of DSP (1/2)

Absence of drift in the filter characteristics

Processing characteristics are fixed, e.g. by binary

coefficients stored in memories

Independent of the external environment and of

parameters such as temperature and device aging

Improved quality level

Quality of processing limited only by economic

considerations

Desired quality level achieved by increasing the

number of bits in data/coefficient representation (SNR

improvement: 6 dB/bit)


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Advantages of DSP (2/2)

Reproducibility

Component tolerances do not affect system

performance with correct operation

No adjustments necessary during fabrication

No realignment needed over lifetime of equipment

Ease adjustment of processor characteristics

Easy to develop and implement adaptive filters,

programmable filters and complementary filters

Time-sharing of processor (multiplexing & modularity)

No loading effect

Realization of certain characteristics not possible or

difficult with analog implementations (e.g., linear phase)


Limitations of DSP

Limited Frequency Range of Operation

Frequency range is technologically limited to values

corresponding to maximum computing capacities (e.g.,

A/D converter) that can be developed and exploited

Digital systems are active devices, thereby

consuming more power and being less reliable

Additional Complexity in the Processing of Analog

Signals

A/D and D/A converters must be introduced, thus adding

complexity to the overall system

Inaccuracy due to finite precision arithmetic


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Continuous-Time Sinusoidal Signals


Properties:

For every fixed value of F, xa(t) is periodic (xa(t + Tp) = xa(t))

Continuous-time sinusoidal signals with distinct frequencies

are themselves distinct

Increasing the frequency F results in an increase in the rate

of oscillation of the signal

( ) cos

cos 2 ,

ax t A t

A Ft t

Representation of Sinusoidal Signals Using

Complex-Conjugate Exponentials


( ) cos

2 2

a

j t j t

x t A t

A Ae e

cos sinje j

Euler identity:

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Discrete-Time Sinusoidal Signals


Properties:

A discrete-time sinusoid is periodic only if its frequency f is a

rational number

Discrete-time sinusoids whose frequencies are separated

by an integer multiple of 2 are identical

The highest rate of oscillation in a discrete-time sinusoid is

attained when = (or = )

( ) cos

cos 2

, 2 2

j n j n

x n A n

A fn

A Ae e n

Discrete-Time Sinusoidal Signals


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Harmonically Related Complex Exponentials


0 02( ) , 0, 1, 2,...

jk t j kF t

ks t e e k

Continuous-time, harmonically related exponentials:

0( ) ( )jk t

a k k k

k k

x t c s t c e

For a continuous-time periodic signal with fundamental

period Tp = 1/F0, its Fourier series expansion is

1 1

2 /

0 0

N Nj kn N

k k k

k k

x n c s n c e

For a discrete-time periodic signal with fundamental period N,

its Fourier series expansion becomes

Analog-to-Digital (A/D) Conversion


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Sampling of Analog Signals (Cont.)


Periodic/Uniform Sampling

Sampling rate (or sampling frequency) Fs = 1/T, where T

is the sampling period (interval)

( ), ax n x nT n

Sampling of Analog Signals (Cont.)


Consider a sinusoidal signal

The discrete signal sampled at Fs can be represented as

( ) cos 2ax t A Ft

2

( ) cos 2 cosa

S

nFx nT A FnT A

F

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Aliasing (1/2)


The phenomenon of a continuous-time signal of higher

freq. acquiring the identity of a sinusoidal sequence of

lower freq. after sampling is called aliasing

An infinite number of continuous-time signals can lead to

the same sequence when sampled periodically

0

0

( ) cos 2

cos 2 / 2

cos 2

oa

S

S

F kFx nT A n

F

A nF F kn

A f n

0( ) cos 2 , where , 1, 2,..a k k Sx t A F t F F kF k

x n

Aliasing (2/2)


Given the samples, draw a sinusoid through the values

cos(0.4 )x n n)4.2cos()4.0cos(

integer an is When

nn

n

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Ambiguity in Sampling


Sample the following three signals at 10 Hz

we obtain g1[n] = g2[n] = g3[n]


Artifacts due to Sampling Aliasing (1/2)

http://upload.wikimedia.org/wikipedia/commons/3/31/Moire_pattern_of_bricks.jpg

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Artifacts due to Sampling Aliasing (2/2)

Sampling Theorem (1/2)


Suppose any analog signal can be represented as a sum of sinusoids of different amplitudes

Thus if Fs 2Fmax= max(F1, F2, …, FN) then the

corresponding digital frequencies of the discrete-time signal

obtained by sampling the parent continuous-time sinusoidal

signal will be in the range −1/2 < fi < 1/2

No Aliasing

On the other hand, if Fs < 2Fmax, the normalized digital angular frequencies may foldover into a lower digital frequency ωi = ⟨2πFi/Fs⟩2π in the range −π < ω < π because of aliasing

1

( ) cos 2N

a i i i

i

x t A Ft

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Sampling Theorem (2/2)


Sampling Theorem

If the highest frequency contained in an analog signal xa(t) if

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

( )a a

n S S

n nx t x g t

F F

sin 2 / 2( )

2 2 / 2a a

n

B t n Bnx t x

B B t n B

If Fs = 2B

where / ( ) ( )a S ax n F x nT x n

sin 2( )

2

Btg t

Bt

Quantization of Continuous-Amplitude

Signals


Quantization: the process of converting a discrete-time

continuous-amplitude signal into a digital signal by

expressing each sample value as a finite number of digits

Quantization error/noise: the error introduced in

representing the continuous-valued signal by a finite set

of discrete value levels

non-invertible process

(many-to-one mapping)

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Quantization & Quantization Error


Example of Quantization


qx n Q x n

q qe n x n x n

0.9 , 0

0. 0

n nx n

n

Quantization:

Quantization Error:

An Example:

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Digital-to-Analog (D/A) Conversion


Zero-Oder Hold Linear Interpolation

Ideal D/A (Sinc Interpolation)

Application Examples of DSP

Cellular Phone

Discrete Multitone Transmission (ADSL)

Digital Camera

Signal Coding & Compression

Signal Enhancement


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Cellular Phone Block Diagram


Cellular Phone Baseband SOC


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Discrete MultiTone Modulation (DMT)

Core technology in the implementation of the

asymmetric digital subscriber line (ADSL) and very-

high-rate DSL (VDSL)

ADSL:

Downstream bit-rate: up to 9 Mb/s

Upstream bit-rate: up to 1 Mb/s

VDSL (e.g., CHT 光世代):

Downstream bit-rate: 13 to 26 Mb/s

Upstream bit-rate: 2 to 3 Mb/s

Distance: less than 1 km

Orthogonal Frequency-Division Multiplexing (OFDM)

for wireless communications (802.11 a/g/n, WiMAX,

LTE, DVB-T/H, etc.)


ADSL modem

DMT

Transmitter

Receiver


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Digital Camera (1/5)



CMOS Imaging Sensor

Increasingly being used in digital cameras

Single chip integration of sensor and other image

processing algorithms needed to generate final image

Can be manufactured at low cost

Less expensive cameras use single sensor with

individual pixels in the sensor covered with either a

red, a green, or a blue optical filter


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DSP-Based Image Processing Algorithms

Bad pixel detection and masking

Color interpolation

Color balancing

Contrast enhancement

False color detection and masking

Image and video compression



Bad pixel detection and masking


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Color Interpolation and Balancing


Signal Coding & Compression

Concerned with efficient digital representation of

audio or visual signal for storage and

transmission to provide maximum quality to the

listener or viewer

Speech coding: ITU-T G.711, G.723.1

Audio coding: MP3

Image coding: JPEG, JPEG-2000

Video Coding: MPEG-1 (VCD), MPEG-2 (DVD),

MPEG-4, H.264, Multi-View Coding (3-D TV)


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Signal Compression Example (1/4)

Original Speech

Data size: 330,780 bytes

• Compressed Speech (GSM 6.10)

Sampled at 22.050 kHz, Data size 16,896 bytes

Compressed speech (Lernout & Hauspie CELP 4.8kbit/s)

Sampled at 8 kHz, Data size 2,302 bytes



Original Music

Audio Format: PCM 16.000 kHz, 16 Bits

(Data size 66206 bytes)

Compressed Music

Audio Format: GSM 6.10, 22.05 kHz

(Data size 9295 bytes)

Courtesy: Dr. A. Spanias


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Original Lena Image

File Size = 256K bytes

Compressed Lena Image

File Size = 13K bytes




Compression Rate: 130:1

JPEG2000(7KB, 5922 bytes) JPEG (7KB, 6220 bytes)

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Applications: Signal Enhancement

Purpose: To emphasize specific signal features

to provide maximum quality to the listener or

viewer

For speech signals, algorithms include removal

of background noise or interference

For image or video signals, algorithms include

contrast enhancement, sharpening and noise

removal


Signal Enhancement Examples (1/4)

Noisy speech signal

(10% impulse noise)

Noise removed speech


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Original image and its contrast enhanced version

Original Enhanced


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Noisy image & denoised image


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Chapter 1 Introduction - ee.nthu.edu.tw