Review of Digital Signal Processing SMJE...
Transcript of Review of Digital Signal Processing SMJE...
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)
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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)
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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