Week 1Introduction of Digital Signal Processing

with the review of SMJE 2053 “Circuits & Signals”for Filter Design

Week 1Introduction of Digital Signal Processing

with the review of SMJE 2053 “Circuits & Signals”for Filter Design

1) Signals, Systems, and DSP

2) DSP system configuration

3) Applications of DAP

4) Review of Discrete-time systems

1) Signals, Systems, and DSP

2) DSP system configuration

3) Applications of DAP

4) Review of Discrete-time systems

1

SMJE3163 DSP2016_Week1-04

►Signal

►Signal Processing

►Digital Signal Processing:

►Benefits

of DSP

►Signal

►Signal Processing

►Digital Signal Processing:

►Benefits

of DSP2

Signals convey some information through its physical variation pattern.

concerns with the representation, transformation, manipulation of signals and the extraction of the information containing signals.

Processing of analog signals by digital computer/ devices.

Flexibility, High Reliability, Consistency

Signals convey some information through its physical variation pattern.

concerns with the representation, transformation, manipulation of signals and the extraction of the information containing signals.

Processing of analog signals by digital computer/ devices.

Flexibility, High Reliability, Consistency

1) Signals, Systems, and DSP

物理量の変化パターンを通じて

情報を担うもの

Speech sound waveforms

/a, あ/ sound /I, い/ sound

Types of Signals

Time

Continuous-Time Discrete-Time

Am

pli

tude Continuous-

Value

Discrete-Value

Types of Signals

Time

Continuous-Time Discrete-Time

Am

pli

tude Continuous-

ValueAnalog Signal

Discrete-Time

Signal

Discrete-Value Digital Signal

Digital

ComputerADC DAC

analog

signalanalog

signal

DSP System TYPE 1: Manipulate input signal to generate output signal

2) DSP system Configurations

ADC: Analog DigitalConverter

DAC:Digital Analog Converter

analog signal

n

* During the value is hold constant, the value is quantized and converted it to binary code. [Tan-Ch.2]

series of binary codes

A-D Converter

*

time in

dex

-4Δ

-3Δ

-2Δ

-Δ

Δ

2Δ

3Δ

3Δ/2Δ/2

Δ：quantization step size

5Δ/2

-Δ/2-3Δ/2-5Δ/2-7Δ/2

Quantization and Binary Code Generation

Characteristics of bipolar quantization (rounding)

111

110

101

100

011

010

001

000

Binary code

x

Q(x)

analog

signal

n

series of binary codes

DAC interpolates the values between the sampled values.

Stair case reconstruction followed by the smoothing filter is most commonly used

Smooth-ingfilter

zero-order-hold

Decoding

DA Conversion

00001010001011011000100101001110

..

Binary Code

D-A Converter

Discrete-Time System

“Sample-by sample” processing:

The output sample y(n) is computed, when the input

x(n) is available at the input, and the computation has to

be completed before the next sample x(n+1) appears at

the input port.

Input x(n) Output y(n)Discrete-Time

System

Digital

Computer

A/D

Converter

Processed

results

analog

signal

Send to

/ Display

/ Pattern

Recognition

etc.

Other DSP System Configurations

DSP TYPE 2: Manipulating signal to get (estimate) parameters which characterize the input signal. No DAC device.

Digital

ComputerDAC

analog

signal

DSP TYPE 3: Computer generates analog output signalNo ADC system, for example ?

Tasks of DSP:

►Filtering/Separation

►Enhancement of target signal

►Noise Cancellation

►Spectrum Estimation

►Characterization of Signals for pattern recognition

►Data Reduction without losing signal quality

Tasks of DSP:

►Filtering/Separation

►Enhancement of target signal

►Noise Cancellation

►Spectrum Estimation

►Characterization of Signals for pattern recognition

►Data Reduction without losing signal quality 12

3) Applications of DSP

Fields of DSP ApplicationFields of DSP Application

►Consumer/Commercial/Multi-Media

Audio-Speech/ Image/ Video Processing

►Communication System

►Medical/Bio Science

►Industrial Use

►Scientific Analysis

►Military/Radar Sonar etc.

►Space

►Consumer/Commercial/Multi-Media

Audio-Speech/ Image/ Video Processing

►Communication System

►Medical/Bio Science

►Industrial Use

►Scientific Analysis

►Military/Radar Sonar etc.

►Space13

Fields of DSP ApplicationFields of DSP ApplicationFields of DSP ApplicationFields of DSP Application

Microphone array system

Experiment movie

Mixed signal

Separated voices by Microphone Array System

Separation of Speech Signals

Processing by Microphone array

（Keio Univ. Hamada ＆ Nakazawa Labs 2012)

Microphone array

Mobile Robot

Specific Speaker Tracking under Noisy environment

Super-resolution image reconstruction

Real sequence of images

Acquired 20 Images： 176×110

Super Resolution Image 352×220

4) Review: Discrete-time Systems

4-1) Description of LTI discrete-time systems

► Difference equation,

► Block diagram

► Impulse (Unit-Sample) Response

► System function

4-2) Frequency Response

► Amplitude and Phase characteristics

LTI: Linear Time-Invariant

4-1) Description of LTI discrete-time systems

► Difference equation,

► Block diagram

► Impulse (Unit-Sample) Response

► System function

4-2) Frequency Response

► Amplitude and Phase characteristics

LTI: Linear Time-Invariant

4-1 Description of discrete-time systems

►Difference equation (1st order system)

4-1 Description of discrete-time systems

►Difference equation (1st order system)

Ex. 1) Non-recursive System

𝑦 𝑛 = 𝑏0𝑥 𝑛 + 𝑏1𝑥 𝑛 − 1 + 𝑏2𝑥 𝑛 − 2 (1.1)

Ex. 2) Recursive System

𝑦 𝑛 = 𝑎𝑦 𝑛 − 1 + 𝑏0𝑥 𝑛 + 𝑏1𝑥 𝑛 − 1 (1.2)

Input signal

(sequence)

𝑥(n)

Output Signal

y(n)Discrete-Time

System

►Block diagram:

Three basic operations:

Unit delay, Multiplier (Gain), Adder (Sum)

►Block diagram:

Three basic operations:

Unit delay, Multiplier (Gain), Adder (Sum)

𝑏𝑧−1 +

(a) unit delay (b) Multiplier

(Gain)

(c) Adder

(Sum)

𝑏Σ

𝑏0 𝑏1 𝑏2

𝑧−1 𝑧−1

++

𝑥(n)

𝑥(n-1) 𝑥(n-2)

𝑦(n)

Block diagram of system (1.1)

𝑏0

𝑏1 𝑎

𝑧−1 𝑧−1

++𝑦(n)

𝑦(n-1)

𝑥(n)

𝑥(n-1)

Block diagram of system (1.2)

►Impulse (Unit-Sample) Response

characterizes discrete-time system completely. This means that the output for any input can be computed by using solely impulse response.

►Impulse (Unit-Sample) Response

characterizes discrete-time system completely. This means that the output for any input can be computed by using solely impulse response.

Unit impulse function

𝛿 𝑛 = 1 , 𝑛 = 0

0 , 𝑛 ≠ 0

Exercise 1 Compute the impulse response sequence of systems of Ex.1) and

Ex.2) respectively, and then find the length of impulse responses.

unit impulse

𝛿(n)

Impulse response

h(n)

Discrete-Time

System

The Impulse (Unit-Sample) Response

Copyright © 2014 McGraw-Hill Education. Permission required for reproduction or display

Due to linearity property, if one knows the unit-sample response of an LTI system one can compute system’s response to any input.

𝑛 = 0

𝑛

𝑛 = 0

𝑛

𝛿 𝑛 :unit − sample

ℎ(𝑛): Impulse (unit-sample) response

Q.1 Find the impulse Response from the difference eq. (1.1)

n<0 h(n)=0,

n=0 h(0)=b0, n=1 h(1)=b1,

n=2 h(2)=b2, n≧3 h(n)=0

Since the length of above response is finite, this type of system

is referred to as finite impulse response (FIR) system/filter.

Q.2 Find the impulse Response from the difference eq. (1.2)

n<0 h(n)=0,

n=0 h(0)=b0, n=1 h(1)=a b0 + b1

n=2 h(2)=a(ab0 + b1 ), n=3 h(3)=a2(a b0 + b1 ),

……. endless,

Since the length of this impulse is finite, this type of system is

referred to as infinite impulse response (IIR) system/filter.

►System/Transfer function:

The ratio of input and output in the z-domain

►System/Transfer function:

The ratio of input and output in the z-domain

z-transform:

𝑋 𝑧 =

𝑛=−∞

𝑛=∞

𝑥 𝑛 𝑧−𝑛

System function:

𝐻 𝑧 =𝑌 𝑧

𝑋 𝑧

System functions of Ex.1) and Ex.2) are given by:

Ex1) 𝐻 𝑧 = 𝑏0 + 𝑏1𝑧−1 + 𝑏2𝑧

−2 (1.3)

Ex2) 𝐻 𝑧 =𝑏0+𝑏1𝑧

−1

1−𝑎𝑧−1(1.4)

4-2 Frequency Response

►Definition:

The complex-valued gain 𝐻 𝑒𝑗Ω of the system for the

input 𝑒𝑗Ω𝑛

4-2 Frequency Response

►Definition:

The complex-valued gain 𝐻 𝑒𝑗Ω of the system for the

input 𝑒𝑗Ω𝑛

𝑒𝑗Ω𝑛 Discrete-Time

System𝐻 𝑒𝑗Ω 𝑒𝑗Ω𝑛

Frequency response can be found from the system function 𝐻 𝑧 by

𝐻 𝑒𝑗Ω = 𝐻 𝑧 𝑧=𝑒𝑗Ω (1.5)

Q. 3

Compute the frequency response of system (1.1) with the parameters, b0=1, b1=2, b2=1.

Solution:Substituting the parameters into (1.3) gives

𝐻 𝑧 = 1 + 2𝑧−1 + 𝑧−2 (1.7)

and

𝐻 𝑒𝑗Ω = 1 + 2𝑒−𝑗Ω + 𝑒−𝑗2Ω (1.8)

= 𝑒−𝑗Ω 𝑒𝑗Ω +2 + 𝑒−𝑗Ω = 𝑒−𝑗Ω 2 + 2cosΩ (1.9)

The polar form of frequency response gives the magnitude response 𝐴 Ω and the phase response 𝜃 Ω .

𝐻 𝑒𝑗Ω = 𝐴 Ω 𝑒𝑗𝜃 Ω (1.6)

The polar form of frequency response gives the magnitude response 𝐴 Ω and the phase response 𝜃 Ω .

𝐻 𝑒𝑗Ω = 𝐴 Ω 𝑒𝑗𝜃 Ω (1.6)

13-30

18.3 The Frequency Response 𝑯(𝝎)

𝑯 𝝎 : frequency response of the system.

■ DTFT of the unit-sample response h(n) of

the system,

■It can be found from the system function H(z).

Definition

𝑥 𝑛 = 𝑒𝑗𝜔𝑛 𝑦 𝑛 = 𝐻(𝜔)𝑒𝑗𝜔𝑛

LTI system

𝐻(𝜔)

13-31

18.3 The Frequency Response 𝑯(𝑒𝑗Ω)

𝑯 𝝎 : frequency response of the system.

■ DTFT of the unit-sample response h(n) of

the system,

■It can be found from the system function H(z).

Definition

𝑥 𝑛 = 𝑒𝑗Ω𝑛 𝑦 𝑛 = 𝐻(𝑒𝑗Ω)𝑒𝑗Ω𝑛

LTI system

𝐻(𝑒𝑗Ω)

The magnitude of 𝐻 𝑒𝑗Ω is

A Ω = 𝐻 𝑒𝑗Ω = 2 1 + cosΩ (1.10)

, and the phase response of 𝐻 𝑒𝑗Ω is

θ Ω = ∠𝐻 𝑒𝑗Ω = −Ω (1.11)

Q. 4Compute the frequency response of system (1.1) with the parameters, b0= －1, b1=2, b2=－1.

Solution:Substituting the parameters into (1.3) gives

𝐻 𝑧 = −1 + 2𝑧−1 − 𝑧−2 (1.12)

and

𝐻 𝑒𝑗Ω = −1 + 2𝑒−𝑗Ω − 𝑒−𝑗2Ω (1.13)

= 𝑒−𝑗Ω −𝑒𝑗Ω +2 − 𝑒−𝑗Ω

= 𝑒−𝑗Ω 2 − 2cosΩ (1.14)

Magnitude Response

Phase Response[rad]

Ω0

1

2

3

4

5

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0 Ω

System (1.1) with 𝑏0 = 1, 𝑏1 = 2, 𝑏2 = 1

0 π

0

1

2

3

4

5

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Ω

Ω

0 π

MagnitudeResponse

Phase Response[rad]

System (1.1) with 𝑏0 = −1, 𝑏1 = 2, 𝑏2 = −1

-4

-3

-2

-1

0

1

2

3

4

グラフタイトル

● : input sequence● : output sequence

Output Response by Frequency Response Graph

Q. 5 Find a rough output signal of the system (1.1) with 𝑏0 = 1, 𝑏1 = 2, 𝑏2 = 1 when the input is:

𝑥 𝑛 = cos2𝜋

3𝑛

What’s Next (week2 class)?What’s Next (week2 class)?

37

In Week 2 & 3, we focus Filter Design Methods in the discrete-time domain.

►Relationship between characteristics of discrete-time signal processing and overall analog system

Summary of Week 1Summary of Week 1

▶ Course Outline, Weekly Schedule

►We discussed:

what is DSP, what benefits it brings,

Broad fields of DSP application

►Review of Discrete-time System for Filter

Design

▶ Course Outline, Weekly Schedule

►We discussed:

what is DSP, what benefits it brings,

Broad fields of DSP application

►Review of Discrete-time System for Filter

Design

38