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Lecture 12 Outline: FIR Filter Design
l Announcements:l Midterm announcements next week: MT on May 11 in class
l Review of Last Lecture
l Introduction to FIR Filter Design
l Impulse Response Matching
l Frequency Response Matching
l Causal Design and Group Delay
Review of Last Lecturel ASK/PSK Demodulation (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
n(t)
Review ContinuedQuadrature Digital Modulation: MQAM
l Sends different bit streams on the sine and cosine carriersl Baseband modulated signals often has L>2 levels, with L=2l
l More levels for the same TX power leads to smaller noise immunity and hence higher error probability
l Has M=L2 possible values for (m1(kTs), m2(kTs))®log2M bits per symbol time Ts, Data rate is log2M/Ts bps; called MQAM modulation
l 10 Gbps WiFi has 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: log2M 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
Main Points
l FIR filter design entails approximating an ideal discrete or continuous filter with a discrete filter of finite duration
l Impulse response and frequency response matching minimizes time/frequency domain error; have same noncausal designl Optimal filter has M+1 of original discrete-time impulse response valuesl Sharp windowing causes “Gibbs” phenomenon (wiggles)
l Make design causal via a delayl Practical implementation can introduce phase distortion via group delay
l Use windowing to mitigate Gibbs phenomenon (Mon lecture)