Digital Modulation (Part II)M.H. Perrott©2007 Digital Modulation (Part II), Slide 3 Transmit and...
Transcript of Digital Modulation (Part II)M.H. Perrott©2007 Digital Modulation (Part II), Slide 3 Transmit and...
M.H. Perrott©2007 Digital Modulation (Part II), Slide 1
DigitalModulation(Part II)
• Receiver noise vs. intersymbol interference (ISI)• Nyquist Criterion for zero ISI• (Square-Root) Raised Cosine Filter• Complex mixing for frequency offset removal
Copyright © 2007 by M.H. PerrottAll rights reserved.
M.H. Perrott©2007 Digital Modulation (Part II), Slide 2
Review of Digital I/Q Modulation
• Leverage analog communication channel to send discrete-valued symbols– Example: send symbol from set {-3,-1,1,3} on both
I and Q channels each symbol period• At receiver, sample I/Q waveforms every symbol
period– Associate each sampled I/Q value with symbols from
set {-3,-1,1,3} on both I and Q channels
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(f)
Lowpass
H(f)
Receiver Output
t
t
t
Baseband Input
t
SampleTimes
31
-1-3
31
-1-3
31
-1-3
31
-1-3
I
Q
M.H. Perrott©2007 Digital Modulation (Part II), Slide 3
Transmit and Receiver Filters
• Transmit filter examined last time– Tradeoff of transmitted bandwidth vs
intersymbol interference (ISI)• Receive filter examined this time
– Previously assumed to have very wide bandwidth so as not to influence ISI
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(f)
Lowpass
H(f)
Receiver Output
t
t
SampleTimes
31
-1-3
31
-1-3
I
Q
Lowpass
P(f)
Lowpass
P(f)
f0-1
T1T
t
T dq(t)
t
T di(t)
n(t)
y(t)
|Y(f)|
(for one trial)
f
|H(f)||P(f)|
(ZOH)
fTransmit Filter Receive Filter
t
p(t)
(ZOH)
T
Q
00 01 11 10
00
01
11
10
I
Constellation Diagram
M.H. Perrott©2007 Digital Modulation (Part II), Slide 4
Tools for ISI Examination
• Eye diagram– Shows transition behavior
between symbols– ISI causes closing of eye
• Constellation diagram– Shows aggregate placement of sampled
I/Q values– ISI causes spreading of symbol points
I and Q Eye Diagrams
tSampleTimes
p
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(f)
Lowpass
H(f)
Lowpass
P(f)
Lowpass
P(f)
t
T dq(t)
t
T di(t)
n(t)
y(t)
f
|H(f)|
Receive Filter
Q
00 01 11 10
00
01
11
10
I
Constellation Diagram
M.H. Perrott©2007 Digital Modulation (Part II), Slide 5
Impact of Receiver Noise
• Receiver noise adds to desired I/Q signals and causes corruption– Eye diagram reveals closing of eye– Constellation reveals high spread in
symbol points• Key insight: lowering the receive
filter bandwidth improves rejection of noise
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(f)
Lowpass
H(f)
Lowpass
P(f)
Lowpass
P(f)
t
T dq(t)
t
T dq(t)
n(t)
y(t)
Q
00 01 11 10
00
01
11
10
I
f
|H(f)|
f0
|N(f)|
(one trial)
Receive Filter
I and Q Eye Diagrams
tSampleTimes
p
Constellation Diagram
M.H. Perrott©2007 Digital Modulation (Part II), Slide 6
Impact of Lower Receive Filter Bandwidth
• Receiver noise is assumed to be white(i.e., flat spectrum)– Noise is primarily composed
of thermal noise in receive circuits
• Receive filter only passes noise within its passband
How much can we lower thereceive filter bandwidth?
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(f)
Lowpass
H(f)
Lowpass
P(f)
Lowpass
P(f)
t
T dq(t)
t
T dq(t)
n(t)
y(t)
Q
00 01 11 10
00
01
11
10
I
f
|H(f)|
f0
|N(f)|
(one trial)
Receive Filter
I and Q Eye Diagrams
tSampleTimes
p
Constellation Diagram
M.H. Perrott©2007 Digital Modulation (Part II), Slide 7
ISI Versus Noise
• Lowering receive filter bandwidth too much causes ISI to dominate
• Selection of receive filter bandwidth involves a tradeoff between ISI and noise– Bandwidth too high: high noise– Bandwidth too low: high ISI
We need to learn more about ISI
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(f)
Lowpass
H(f)
Lowpass
P(f)
Lowpass
P(f)
t
T dq(t)
t
T dq(t)
n(t)
y(t)
Q
00 01 11 10
00
01
11
10
I
f
|H(f)|
f0
|N(f)|
(one trial)
Receive Filter
I and Q Eye Diagrams
tSampleTimes
p
Constellation Diagram
M.H. Perrott©2007 Digital Modulation (Part II), Slide 8
Joint Transmit/Receive ISI Analysis
• ISI influenced by transmit and receive filter– Define combined filter response G(f) = P(f)H(f)
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(f)
Lowpass
H(f)
Receiver Output
t
t
SampleTimes
31
-1-3
31
-1-3
I
Q
Lowpass
P(f)
Lowpass
P(f)
t
T dq(t)
t
T di(t)
n(t)
y(t)
f
H(f)P(f)
(ZOH)
f
Transmit Filter Receive Filter
t
p(t) (ZOH)
T
G(f)
f
Overall Filter
t
h(t)
t
g(t)
T
*
M.H. Perrott©2007 Digital Modulation (Part II), Slide 9
Viewing Filtering in the Time Domain
• Filtering operation corresponds to convolution in the time domain with impulse response (i.e., g(t) )
• Time domain view allows us to more directly see impact of overall filter on ISI
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(f)
Lowpass
H(f)
Receiver Output
t
t
SampleTimes
31
-1-3
31
-1-3
I
Q
Lowpass
P(f)
Lowpass
P(f)
t
T dq(t)
t
T di(t)
n(t)
y(t)
t
g(t)
T
*
G(f)di(t) ir(t)
t
Tdi(t)
t
ir(t)3
1
-1
-3
M.H. Perrott©2007 Digital Modulation (Part II), Slide 10
Impulse Response and ISI (High Bandwidth)
• Receiver samples I/Q symbols every symbol period– Achievement of zero ISI requires
that each symbol influence only one sample at the filter output
• Issue: we want lower overall filter bandwidth to reduce transmitter spectrum bandwidth and/or lower receiver noise– Causes smoothing of g(t)
t
g(t)
T
*
G(f)di(t) ir(t)
t
Tdi(t)
t
ir(t)3
1
-1
-3SampleTimes
tSampleTimes
Eye Diagram
M.H. Perrott©2007 Digital Modulation (Part II), Slide 11
Impulse Response and ISI (Low Bandwidth)
• Smoothed impulse response has a span longer than one symbol period– Convolution operation reveals
that each symbol impacts filter output at more than one sample value
• Intersymbol interference occurs
SampleTimes
t
g(t)
T
*
G(f)di(t) ir(t)
t
Tdi(t)
t
ir(t)3
1
-1
-3
Eye Diagram
tSampleTimes
M.H. Perrott©2007 Digital Modulation (Part II), Slide 12
A More Direct View of ISI Issue
• Consider impact of just one symbol– Samples at filter output
more clearly show the impact of the one symbol on other sample values
SampleTimes
t
g(t)
T
*
G(f)di(t) ir(t)
t
Tdi(t)
t
ir(t)3
1
-1
-3
Eye Diagram
tSampleTimes
Intersymbol
Interference
M.H. Perrott©2007 Digital Modulation (Part II), Slide 13
The Nyquist Criterion for Zero ISI
• Sample the impulse response of the overall filter at the symbol period
• Resulting samples must have only one non-zero value to achieve zero ISI
Can we design impulse response to span more than one symbol period and stillmeet the Nyquist Criterion for zero ISI?
t
g(t)
T
*
G(f)di(t) ir(t)
t
Tdi(t)
t
ir(t)3
1
-1
-3SampleTimes
tSampleTimes
Eye Diagram
M.H. Perrott©2007 Digital Modulation (Part II), Slide 14
Raised Cosine Filter
• Raised cosine filter achieves low bandwidth and zero ISI– Impulse response spans more than one symbol, but has
only one non-zero sample value– Described by function:
What is the impact of the parameter α?
T
t
g(t)
f
|G(f)|
1/T
α = 1.0 α = 1.0
M.H. Perrott©2007 Digital Modulation (Part II), Slide 15
Impact of α for Raised Cosine Filter
• Parameter α is referred to as the roll-off factor of the filter, where 0 ≤ α ≤ 1
• Smaller values of α lead to– Reduced filter bandwidth– Increased duration of the filter impulse response
• Regardless of the value of α, the raised cosine filter allows achieves zero ISI– Eye diagrams useful to see impact of α
T
t
g(t)
f
|G(f)|
1/T
α = 1.0α = 0.5
α = 1.0α = 0.5
M.H. Perrott©2007 Digital Modulation (Part II), Slide 16
Impact of Large α on Eye Diagram
• Large roll-off factor leads to nice, open eye diagram
• Key observation: achievement of zero ISI requires precise placement of sample times– Error in placement of sample
times leads to substantial ISI!
t
g(t)
T
*
G(f)di(t) ir(t)
t
Tdi(t)
t
ir(t)3
1
-1
-3SampleTimes
tSampleTimes
Eye Diagram
α = 1.0
M.H. Perrott©2007 Digital Modulation (Part II), Slide 17
Impact of Small α on Eye Diagram
• Small roll-off factor reduces the filter bandwidth and still allows zero ISI to be achieved
• Issue: sensitivity to sample time placement is higher than for large α– Receiver complexity must be
higher to insure high accuracy of sample time placement
t
g(t)
T
*
G(f)di(t) ir(t)
t
Tdi(t)
t
ir(t)3
1
-1
-3SampleTimes
tSampleTimes
Eye Diagram
α = 0.5
M.H. Perrott©2007 Digital Modulation (Part II), Slide 18
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(f)
Lowpass
H(f)
Receiver Output
t
t
SampleTimes
31
-1-3
31
-1-3
I
Q
Lowpass
P(f)
Lowpass
P(f)
t
T dq(t)
t
T di(t)
n(t)
y(t)
G(f)di(t) ir(t)
T
t
g(t)
f
|G(f)|
1/T
α = 1.0 α = 1.0
Raised Cosine Filter
tSampleTimes
Eye and Q Eye Diagrams
Transmit and Receiver Filter Design
• Overall response corresponds toG(f) = P(f)H(f)
How do we choose P(f) and H(f) ?
M.H. Perrott©2007 Digital Modulation (Part II), Slide 19
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(f)
Lowpass
H(f)
Receiver Output
t
t
SampleTimes
31
-1-3
31
-1-3
I
Q
Lowpass
P(f)
Lowpass
P(f)
t
T dq(t)
t
T di(t)
n(t)
y(t)
t
p(t), h(t)
f
|P(f)|, |H(f)|
1/T
α = 1.0 α = 1.0
Square-Root Raised Cosine Filter
tSampleTimes
Eye and Q Eye Diagrams
Matched Filter Design
• Setting H(f) = P(f) yields matched filter design– Each filter is chosen to be a Square-
Root Raised Cosine filter– 6.011 will discuss in more detail
M.H. Perrott©2007 Digital Modulation (Part II), Slide 20
Motivation for Complex Mixing
• Issue: is there a convenient way to undo the receiver frequency offset after demodulation by the first set of receive mixers?
• Consider looking at received I/Q signals as a complex signal:
2cos(2π(fo+foff)t)
2sin(2π(fo+foff)t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
y(t) r(t)
rxa(t)
rxb(t)
M.H. Perrott©2007 Digital Modulation (Part II), Slide 21
2cos(2π(fo+foff)t)
2sin(2π(fo+foff)t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
y(t) r(t)
rxa(t)
rxb(t)
i(t)
q(t)
Complex Mixer
e-j2πfofft
Complex Mixer Simplifies Offset Removal
• Simply multiply input by complex exponential– Output will also be a complex signal
M.H. Perrott©2007 Digital Modulation (Part II), Slide 22
2cos(2π(fo+foff)t)
2sin(2π(fo+foff)t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
y(t)
Complex Mixer
cos(2πfofft)
sin(2πfofft)
cos(2πfofft)
r(t)
rxa(t)
rxb(t)
i(t)
q(t)
• Complex mixing achieved with four real mixers– Add/subtract outputs appropriately
Practical Implementation of Complex Mixer
M.H. Perrott©2007 Digital Modulation (Part II), Slide 23
Summary• Receive filter design involves a tradeoff between
noise and ISI– Higher bandwidth leads to higher noise– Lower bandwidth leads to higher ISI
• Zero ISI can be achieved if Nyquist Criterion is met– Choose transmit and receive filters to both be Square-Root
Raised Cosine Filters
• Complex mixing offers a convenient way of removing frequency offset– We will make use of this in Lab 5
• Next lecture: further examine impact of noise in digital modulation