EE 102b: Signal Processing and Linear Systems IIMidterm Review

Signals and Systems

Sampling and Reconstruction vs. Analog-to-Digital and Digital-to-Analog Conversion

l Sampling: converts a continuous-time signal to a continuous-time sampled signal

l Reconstruction: converts a sampled signal to a continuous-time signal.

l Analog-to-digital conversion: converts a continuous-time signal to a discrete-time quantized or unquantized signal

l Digital-to-analog conversion. Converts a discrete-time quantized or unquantized signal to a continuous-time signal.

0 Ts 2Ts 3Ts 4Ts-3Ts -2Ts -Ts

0 1 2 3 4-3 -2 -1

0 Ts 2Ts 3Ts 4Ts-3Ts -2Ts -Ts

0 1 2 3 4-3 -2 -1

Each level canbe representedby 0s and 1s

xs(t)

Sampling

l Sampling (Time):

l Sampling (Frequency): x(t)p(t) X(jw)*P(jw)/(2p)

l Analog-to-Digital Conversation (ADC)l Setting xd[n]=x(nTs) yields Xs(ejW) with W=wTs

0

x(t) =p(t)=ånd(t-nTs)

Xs(jw)

0 0 0

X(jw) =ånd(w-(2pn/Ts))*

0 Ts 2Ts 3Ts 4Ts-3Ts -2Ts -Ts0 Ts 2Ts 3Ts 4Ts-2Ts -Ts-3Ts

2pTs

-2pTs

2pTs

-2pTs

𝟐𝝅𝑻𝒔

1 1/Ts

Reconstructionl Frequency Domain: low-pass filter (Ts=p/W)

l Time Domain: sinc interpolation

l Digital-to-Analog Conversation (DAC)l LPF applied to Xs(ejW) and then converted to

continuous time (w=W/Ts) recovers sampled signal

Xs(jw)

0 2pTs

-2pTs

W-W w

Ts

w

H(jw)

0-2pTs

Xs(jw)

2pTs

Xr(jw)

( ) ( ) åå¥

-¥=

¥

-¥=÷÷ø

öççè

æ -=-=n s

sss

nsssr T

nTtnTxnTthnTxtx sinc)()(

H(jw)

Nyquist Sampling Theoreml A bandlimited signal [-W,W] radians is completely described

by samples every Ts£p/W secs.

l The minimum sampling rate for perfect reconstruction, called the Nyquist rate, is W/p samples/second

l If a bandlimited signal is sampled below its Nyquist rate, distortion (aliasing occurs)

Xs(jw)X(jw)

W-W WW

X(jw)

2W=2p/Ts-2W 00

Quantization

l Divide amplitude range [-A,A] into 2N levels, {-A+kD}, k=0,…2N-1l Map x(t) amplitude at each Ts to closest level, yields xQ(nTs)=xQ[n]l Convert k to its binary representation (N bits); converts xQ[n] to bits

-A

A

-A+D-A+2D

-A+kD

……

Ts 2Ts …0

xQ(nTs)x(t)

Continuous-TimeUnquantized

x(t) nTt =Anti-AliasingLowpass Filter

Bandlimits x(t) toprevent aliasing

Sampler

Quantizer

Discrete-TimeUnquantized

xd[n] = x(t)| t = nT

Discrete-TimeQuantized Representation

of xd[n] …

Digital Storage,Transmission,

Signal Processing, …

Discrete-TimeQuantized y[n] Reconstruction

SystemContinuous-Time

y(t)

Analog to Bits and Back

Continuous-TimeUnquantized

x(t) nTt =Anti-AliasingLowpass Filter

Bandlimits x(t) toprevent aliasing

Sampler

Quantizer

Discrete-TimeUnquantized

xd[n] = x(t)| t = nT

Discrete-TimeQuantized Representation

of xd[n] …

Digital Storage,Transmission,

Signal Processing, …

Discrete-TimeQuantized y[n] Reconstruction

SystemContinuous-Time

y(t)

Analog to Bits

Bits to Analog

0100100110010010000100001000100111…

0100100110010010000100001000100111…

ADC

DAC

Sampling with Zero-Order Hold

l Ideal sampling not possible in practicel In practice, ADC uses zero-order

hold to produce xd[n] from x0(t)l Reconstruction of x(t) from x0(t) removes h0(t) distortion

l Multiplication in frequency domain by P 𝝎𝝎𝒔

/H0(jw)

l Also use zero-order hold for reconstruction in practice

R1

R2

xd(t) h0(t)1

Ts

x0(t)Hr(jw)

xr(t)=x(t)DiscreteTo

Continuous

xd[n]=x(nTs)

xs(t)ånd(t-nTs)

0 Ts 2Ts 3Ts 4Ts-3Ts -2Ts -Ts

x(t)´

xs(t) h0(t)1

Ts

x0(t) x0(t)xd[n] (Ts=1)

X(jw)

Xs(jw) |H0(jw)|

|X0(jw)|

|Hr(jw)|

x(t)´

xs(t) h0(t)1

T

x0(t)

ånd(t-nTs)

Hr(jw)xr(t)=x(t)

Ho(jw)Hr(jw)=T&P 𝝎𝝎𝒔

T

Zero-order hold model

l Inserts L-1 zeros between each xd[n] value to get upsampledsignal xe[n]

l Compresses Xd(ejW) by L in W domain and repeats it every 2p/L; So Xe(ejW) is periodic every 2p/L

l Leads to less stringent reconstruction filter design than ideal LPF: zero-order hold often used

Discrete-Time Upsampling

0 1 2 3 4 0 L 2L 3L 4L

xd[n] xe[n]

0 p-pW

Xd(ejW)

WW-WW

WW=WTs

2p-2p W

Xe(ejW)

WW-WW

L L

… …

2pL

-2pL

xe[n]UpsampleBy L (­L)

xd[n]=x(nTs)

0w

X(jw)

W-W

Reconstruction of Upsampled Signal

l Pass through an ideal LPF Hi(ejW) to get xi[n] l xi[n]=ẋe[n]=x(nTs/L) if x(t) originally sampled at Nyquist rate (Ts<p/W)l Relaxes DAC filter requirements (approximate LPF/zero-order hold

reconstructs x(t) from xi[n]); better filter to reconstruct from xd[n] needed)

0p-p W

xd[n]=x(nTs)ÛXd(ejW)

WW-WW

Hi(ejW)

2p-2p W

Xe(ejW)

WW-WW

L L

0 L 2L 3L 4L

xe[n]

0 1 2 3 4

xd[n]

xi[n]xe[n]UpsampleBy L

xd[n]=x(nTs)

2pL

-2pL

0 L 2L 3L 4L

xi[n]

Xi(ejW) Û xi[n]Xi(ejW)

x(t)DAC

Reconstruct x(t) from xi[n] by passing it through a DAC

… …

More stringent LPF than for Xd(ejW) Less stringent analog LPFthan to reconstruct from xd[n]

=ẋe[n]

xi[n]=ẋe[n]=x(nTs/L) if Ts<p/W

DiscreteTo Cts Ha(jw)

Ha(ejW)

Ha(ejW)

xi[n] x(t)xr(t)≠x(t)

xd[n]

0 2p-2p W

Xd(ejW)W=wTs

0 2pTs

-2pTs

w

Xs(jw)Ts

0w

X(jw)

W-W

Ts/L

0 2p-2p W

Xe(ejW)W=wTs/L

0w

X(jw)

W-W 0 2pLTs

-2pLTs

w

Xs(jw) .

2p-2p 2p/L-2p/L W

Xe(ejW)

… …

Xi(ejW)=Lrect[WL/(2p)]

��𝑒 𝑒*+,-// 𝐿rect 56789: =𝑋; 𝑒*+,-// 𝐿rect 5678

9:

=𝑋𝑒 𝑒*+,- 𝐿rect 56789: =𝑋𝑒 𝑒*< 𝐿rect =8

9: =𝑋𝑖 𝑒*<

��𝑒 𝑛 ≜ 𝑥 B,7/

⇔𝑋-

678 𝑗𝜔

⇒��𝑒 𝑒*< = ��𝑒

𝑒*+,-// =��𝑒 𝑒*+,-// 𝐿rect 56789:

0 L 2L 3L 4L

xe[n]

0 1 2 3 4

xd[n]=Xe(ejW)

xi[n]=ẋe[n]=x(nTs/L)if Ts<p/W

Proof that xi[n]=ẋe[n]«Xi(ejW) =Xe(ejW)

Digital Downsampling:Fourier Transform and Reconstruction

l Digital Downsamplingl Removes samples of x(nTs) for n≠MTsl Used under storage/comm. constraints

l Repeats Xd(ejW) every 2p/M and scales W axis by Ml This results in a periodic signal Xc(ejW) every 2pl Introduces aliasing if Xd(ejW) bandwidth exceeds p/Ml Can prefilter Xd(ejW) by LPF with bandwith p/M prior to

downsampling to avoid downsample aliasing

xc[n]DownsampleBy M

xd[n]=x(nTs)

0 1 2 3 40 123 …

0 p/M-p/M W’

Xd(ejW’)

0 p-p

Xc(ejW)

-2p 2p

……0 p/M-p/M W’

Xd(ejW’)

0 p-p

Xc(ejW)

-2p 2p

……

W=MW’W=MW’

𝑋𝑐 𝑒*< = JK L 𝑋𝑑 𝑒*(<OPQR)/KKOJ

RTU

xd[n]

0123… 0 2p-2p -p/M p/M W

Xd(ejW)

(9)

0 2p-2p W

Xd(ejW)W=wTs

0 2pTs

-2pTs

w

Xs(jw)Ts

0w

X(jw)

W-W

𝑋𝑑 𝑒*< = J,-∑ 𝑋 𝑗 =W9:X

67YZTOY

MTs

0 2p-2p W

Xc(ejW)W=wMTs

0w

X(jw)

W-W 0 2pMTs

-2pMTs

w

Xs(jw) 𝑋𝐶 𝑒*< = JK,-

∑ 𝑋 𝑗 =W9:\]67

Y^TOY

𝑋𝐶 𝑒*< =1𝑀𝑇𝑠

L 𝑋 𝑗 <OPQ^K,-

=1𝑀 L

1𝑇𝑠

L 𝑋 𝑗 <OPQ(ZKcR)K,-

Y

ZTOY

KOJ

RTU

= JK L 𝑋𝑑 𝑒*(<OPQR)/KKOJ

RTU

Y

^TOY

𝑙 = 𝑘𝑀 +𝑚:L =L L Y

ZTOY

KOJ

RTU

Y

^TOY

0 1 2 3 4

xc[n]=xd[nM]

0 2p-2p p-p W

Xc(ejW)

Repeats Xd(ejW) every 2p/M and scales W axis by M

Communication System Block Diagram

l Modulator (Transmitter) converts message signal or bits into format appropriate for channel transmission (analog signal).

l Channel introduces distortion, noise, and interference.

l Demodulator (Receiver) decodes received signal back to message signal or bits.

l Focus on modulators with s(t) at a carrier frequency wc. Allows allocation of orthogonal frequency channels to different users

ChannelDemodulator

(Receiver)Modulator

(Transmitter)

)(ts )(ˆ ts)(ˆ...ˆˆ21

tmbb)(:signal analog

...:bits 21

tmbb

Amplitude ModulationDSBSC and SSB

l Double sideband suppressed carrier (DSBSC) l Modulated signal is s(t)=m(t)cos(wct)l Signal bandwidth (bandwidth occupied in positive frequencies) is 2W

l Redundant information: can either transmit upper sidebands (USB) only or lower sidebands (LSB) only and recover m(t)l Single sideband modulation (SSB); uses 50% less bandwidth (less $$$)

l Demodulator for DSBSC/SSB: multiply by cos(wct) and LPF

))](())(([5.)cos()()( ccc jMjMttmts wwwww ++-Û=)( wjS)( wjM

LSB

USB USB

W-W w wwc-wc

W 2W

wc-wc X

cos(wct)s(t)

2wc0-2wc

AM Radio

l Broadcast AM has s(t)=[1+kam(t)]cos(wct) with [1+kam(t)]>0l Constant carrier cos(wct) carriers no information; wasteful of powerl Can recover m(t) with envelope detector (diode, resistors, capacitor)l Modulated signal has twice bandwidth W of m(t), same as DSBSC

m(t) X

cos(wct)

+

A s(t)=[A+kam(t)]coswct

1/(2pwc)<<RC<<1/(2pW)

ka

1

)( wjM

W-W

ka

wc-wc

)( wjM

W-W

Quadrature Modulation

DSBSCDemod

DSBSCDemod

m1(t)cos(wct)+m2(t)sin(wct)

LPF

LPF

-90o

cos(wct)

sin(wct)

Sends two info. signals on the cosine and sine carriers

m1(t)

m2(t)

Digital Communication System Block Diagram

l Channel is a physical entity (wire, cable, wireless channel, string)l Cannot send a complex signal over a physical channel: s(t) must be real

l S(jw)=S*(-jw): s(t) real/even ® S(jw) real/even; s(t) real/odd ® S(jw) imaginary/oddl Often write s(t) in terms of in-phase/quadrature components: s(t)=sI(t)cos(wct)-sQ(t)sin(wct)

ADCAnalogSource

AnalogSink DAC

Compression

Decompression

Error-CorrectionEncoding

Error-CorrectionDecoding

BasebandModulation

BasebandDemodulation

PassbandModulation

PassbandDemodulation

Channel

ModulatedWaveform

s(t)

ŝ(t): Corrupted

Copyof s(t)

CompressedSource Bits

EncodedBits

BasebandWaveform

m(t)

DemodulatedWaveform𝑚i (t)

DecodedCompressedSource Bits

DetectedEncoded

Bits

convertscontinuous-time

to bits

Removes redundancy introducescontrolled

redundancy

binary ormulti-level

shifts waveformto carrierfrequency

propagates signalbut adds

distortion, noise& interference

shifts waveformto baseband

compareswaveform tothresholds

to detect bits

correctserrors in

detected bits

restoressource

redundancy

convertsbits to

continuous-time Analog System

Digital SourceBits

Bits

Digital Sink Bits

l Baseband digital modulation converts bits into analog signals y(t) (bits encoded in amplitude)

l Pulse shaping (optional topic)l Instead of the rect function, other pulse shapes usedl Improves bandwidth properties and timing recoveryl Explored in extra credit Matlab problem

Baseband Digital Modulation

1 0 1 1 0 1 0 1 1 0On-Off Polar

t tTb

)()()(*)()()( bk

kbk

k kTtatxfortrecttxkTtrectatm -==-= åå¥

-¥=

¥

-¥=

d

m(t)m(t)A A

-A

l Changes amplitude (ASK), phase (PSK), or frequency (FSK, no covered) of carrier relative to bits

l We use baseband digital modulation as information signal m(t) to encode bits, i.e. m(t) is on-off or polar

l Passband digital modulation for ASK/PSK is a special case of DSBSCl For m(t) on/off (ASK) or polar (PSK), modulated signal is

Passband Digital Modulation

)cos()()cos()()( tkTtrectattmts cbk

kc ww úû

ùêë

é -== å¥

-¥=

ASK and PSK

l Amplitude Shift Keying (ASK)

l Phase Shift Keying (PSK)

îíì

==

==)"0("0)(0)"1(")()cos(

)cos()()(b

bcc nTm

AnTmtAttmts

ww

1 0 1 1

AM Modulation

AM Modulation

m(t)

m(t)

îíì

-=+=

==)"0(")()cos()"1(")()cos(

)cos()()(AnTmtAAnTmtA

ttmtsbc

bcc pw

ww

1 0 1 1

Assumes carrier phase f=0, otherwise need phase recovery of f in receiver

A

-A

A

-A

ASK/PSK Demodulationl Similar to AM demodulation, but only need to choose

between one of two values (need coherent detection)

l Decision device determines which of R0 or R1 that R(nTb) is closest tol For ASK, R0=0, R1=A, For PSK, R0=-A, R1=Al Noise immunity DN is half the distance between R0 and R1

l Bit errors occur when noise exceeds this immunity

s(t) ´

cos(wct+f)

ò ×bT

b

dtT 0

)(2

nTb

Decision Device

“1” or “0” r(nTb)

R0

R1

a0

r(nTb)

r(nTb)+N

Integrator (LPF)

DN

Quadrature Digital Modulation: MQAM

l Sends different bit streams on the sine and cosine carriersl Baseband modulated signals can have L>2 levels

l More levels for the same TX power leads to smaller noise immunity and hence higher error probability:

l Sends 2log2(L)=M bits per symbol time Ts, Data rate is M/Ts bpsl Called MQAM modulation: 10 Gbps WiFi: 1024-QAM (10 bits/10-9 secs)

● L=32 levels

10 11 00 01

t

mi(t)m1(t)cos(wct)+m2(t)sin(wct)+

n(t)

X

-90o

cos(wct)

sin(wct)

X

ò ×bT

b

dtT 0

)(2

ò ×bT

b

dtT 0

)(2

Decision Device

“1” or “0”

R0

R1

a0

rI(nTb)+NI

Decision Device

“1” or “0”

R0

R1

a0

rQ(nTb)+NQ

A

-A

A/3

-A/3

Ts

Data rate: log2L bits/TsTs is called the symbol time

Introduction to FIR Filter Design

l Signal processing today done digitallyl Cheaper, more reliable, more energy-efficient, smaller

l Discrete time filters in practice must have a finite impulse response: h[n]=0, |n|>M/2l Otherwise processing takes infinite time

l FIR filter design typically entails approximating an ideal (IIR) filter with an FIR filterl Ideal filters include low-pass, bandpass, high-passl Might also use to approximate continuous-time filter

l We focus on two approximation methodsl Impulse response and filter response matchingl Both lead to the same filter design

Impulse Response Matchingl Given a desired (noncausal, IIR) filter response hd[n]

l Objective: Find FIR approximation ha[n]: ha[n]=0 for |n|>M/2 to minimize error of time impulse response

l By inspection, optimal (noncausal) approximation is

[ ] ( )W« jdd eHnh

[ ] [ ] [ ] [ ] [ ] 2/||,0][since,

2

2

2

22 Mnnhnhnhnhnhnh aMn

dMn

adn

ad >=+-=-= ååå>£

¥

-¥=

eDoesn’t depend on ha[n]

[ ] [ ]îíì

>£

=2/02/

MnMnnh

nh da

Exhibits Gibbs phenomenonfrom sharp time-windowing

W

( )Wja eH

p- p0

Frequency Response Matchingl Given a desired frequency response Hd(ejW)

l Objective: Find FIR approximation ha[n]: ha[n]=0 for |n|>M/2 that minimizes error of freq. response

l Set and

l By Parseval’s identity:l Time-domain error and frequency-domain error equall Optimal filter same as in impulse response matching

( ) ( )ò-

WW W-=p

pp

e deHeH ja

jd

2

21

[ ] ( )òå-

W¥

-¥=

W=p

ppdeXnx j

n

22

21

[ ] [ ] [ ]nhnhnx ad -= ( ) ( ) ( )WWW -= ja

jd

j eHeHeXand

[ ] [ ]îíì

>£

=2/02/

MnMnnh

nh da 2/||)(

21][ MndeHnh j

da £W= W

-òp

pp

Causal Design and Group Delayl Can make ha[n] causal by adding delay of M/2l Leads to causal FIR filter design

l If ÐHa(ejW) constant, ÐH(ejW) linear in W with slope -.5Ml Most filter implementations do not have linear

phase, corresponding to a constant delay for all W. l Group delay defined as

l Constant for linear phase filtersl Piecewise constant for piecewise linear phase filtersl Nonconstant group delay introduces phase distortion

relative to an ideal filter

[ ] úûù

êëé -=

2Mnhnh a ( ) [ ] W-

=

W å= jnM

n

j enheH0

( ) ( )WW-W = ja

Mjj eHeeH 2

)( WÐW¶¶

- jeH

Art and Science of Windowingl Window design is created as an alternative to the

sharp time-windowing in ha[n]l Used to mitigate Gibbs phenomenonl Window function (w[n]=0, |n|>M/2) given by

l Windowed noncausal FIR design:

l Frequency response smooths Gibbs in Ha(ejW)

l Design often trades “wiggles” in main vs. sidelobesl Hamming smooths out wiggles from rectangular windowl Introduces more distortion at transition frequencies than rectangle

[ ] ( )W« jeWnw

[ ] [ ] [ ] [ ] [ ]nhnwnhnwnh daw ×=×=

( ) ( ) ( )( ) qp

qp

p

q deHeWeH jd

jjw

-W

-

W ò=21

Typical Window Designs

-5 0 50

0.5

1

n

w[n]

boxcar(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

triang(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

bartlett(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

boxcar(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

triang(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

bartlett(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

hann(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

hanning(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

hamming(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

hann(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

hanning(M+1), M = 8

-5 0 50

0.5

1

n

w[n]

hamming(M+1), M = 8

0 0.5 1 1.5 2 2.5 3-70

-60

-50

-40

-30

-20

-10

0

W

20 lo

g 10|W

(ej W

)|

M = 16

BoxcarTriangular

0 0.5 1 1.5 2 2.5 3-70

-60

-50

-40

-30

-20

-10

0

W

20 lo

g 10|W

(ej W

)|

M = 16

HammingHanning

0 0.5 1 1.5 2 2.5 3

-0.2

0

0.2

0.4

0.6

0.8

1

W

W(ej W

)

M = 16

BoxcarTriangular

0 0.5 1 1.5 2 2.5 3

-0.2

0

0.2

0.4

0.6

0.8

1

W

W(ej W

)

M = 16

HammingHanning

l We are given a desired response hd[n] which is generally noncausal and IIR l Examples are ideal low-pass, bandpass, highpass filtersl May be derived from a continuous-time filter

l Choose a filter duration M+1 for M evenl Larger M entails more complexity/delay, less approximation error e

l Design a length M+1 window function w[n], real and even, to mitigate Gibbs while keeping good approximation to hd[n]

l Calculate the noncausal FIR approximation ha[n]

l Calculate the noncausal windowed FIR approximation hw[n]

l Add delay of M/2 to hw[n] to get the causal FIR filter h[n]

Summary of FIR Design

FIR Realization: Direct Form

l Consists of M delay elements and M+1 multipliersl Can introduce different delays at different freq. components of x[n]l Will discuss more when we cover z transforms

l Efficient implementation using Discrete-Fourier Transform (DFT)

M

[ ]nx

]0[h ]1[h ]2[h ]1[ -Mh ][Mh

++ + +

…

…D D D

[ ]1-nx [ ]2-nx [ ]Mnx -[ ]1+-Mnx[ ]nx

[ ] [ ] [ ]å=

-=M

kknxkhny

0

Main Points

l Sampling and reconstruction bridges analog and digital worlds

l Upsampling and downsampling ease implementation requirements

l Analog and digital communications allows transmission of information signals through the airwaves

l FIR filter design approximates perfect filters with a design tailored to a set of engineering tradeoffs.