L12-digital

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From AM Radio to
Digital I/Q Modulation
COS 463: Wireless Networks
Lecture 12
Kyle Jamieson
[Parts adapted from H. Balakrishnan, M. Perrott, C. Terman]
1. Analog I/Q modulation
2. Discrete-time processing of continuous signals
– The Digital Abstraction
– Quantization: Discretizing values
– Sequences: Discretizing time
3. Digital I/Q Modulation
2
Roadmap
• Multiplication (i.e.mixing) operation shifts in frequency
• Low-pass filtering passes only the desired baseband signal at the receiver
• Works with cos or sin carrier signal
3
Review: AM Modulation & Demodulation
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 2
Transmitter Output Receiver Output
2cos(2πfot)
y(t)x(t)
2cos(2πfot)
y(t) r(t)
Lowpass
Filter
w(t)
H(j2πf)
AM Modulation and Demodulation
• Multiplication (i.e., mixing) operation shifts in 
frequency
– Also creates undesired high frequency components at 
receiver
• Lowpass filtering passes only the desired baseband
signal at receiver
What can go wrong here?
Transmitter: Receiver:
• Suppose receiver uses sine, instead of cosine: no output
sin(2πf0t)�cos(2πf0t) = ½ sin(4πf0t)
• Need to synchronize phase of transmitter and receiver local oscillators
(coherent demodulation)
4
Impact of a 90º phase shift
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 3
Transmitter Output Receiver Output
2cos(2πfot)
y(t)x(t)
2sin(2πfot)
y(t) r(t)
Lowpass
Filter
w(t)
H(j2πf)
Impact of 90 Degree Phase Shift
• If receiver cosine wave turns into a sine wave, we 
suddenly receive no baseband signal!
– We apparently need to synchronize the phase of the 
transmitter and receiver local oscillators
• This is called coherent demodulation
• Some key questions:
– How do we analyze this issue?
– What would be the impact of a small frequency offset?
∿∿
Local 
oscillator
Transmitter: Receiver:
• Transmitter and receiver oscillators phase-synchronized
• Demodulated copies add constructively at baseband (close to f = 0)
5
Coherent Demodulation:
Frequency-domain analysis
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 4
f
-fo fo0
1 1
f
X(j2πf)
0
f
-fo fo0
Y(j2πf)
Transmitter Output
2cos(2πfot)
y(t)x(t)
2cos(2πfot)
y(t) r(t)
Lowpass
Filter
w(t)
H(j2πf) Receiver Output
f
-fo fo0
W(j2πf)
2fo-2fo
H(j2πf)
f
-fo fo0
1 1
Frequency Domain Analysis
• When transmitter and receiver local oscillators are 
matched in phase:
– Demodulated signal constructively adds at baseband
∿ ∿
6
90º Phase Shift:
Frequency-domain analysis
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 5
f
-fo fo0
2fo
-2fo
W(j2πf)
f
-f1
f1
0
j
-j
f
-fo fo0
1 1
f
X(j2πf)
0
f
-fo fo0
Y(j2πf)
Transmitter Output
2cos(2πfot)
y(t)x(t)
2sin(2πfot)
y(t) r(t)
Lowpass
Filter
w(t)
H(j2πf) Receiver Output
H(j2πf)
= 0
Impact of 90 Degree Phase Shift
What would happen with a small frequency offset?
• When transmitter and receiver local oscillators are 
90 degree offset in phase:
– Demodulated signal destructively adds at baseband
• Transmitter, receiver oscillators are offset in phase by 90º 
• Demodulated copies add destructively at baseband (close to f = 0)
– Zero output from receiver
• But: Opportunity to create two separate channels!
7
In-Phase/Quadrature Modulation
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 7
Receiver Output
2cos(2πfot)
2sin(2πfot)
y(t)
ir(t)
qr(t)
f
f
f
-fo fo0
1 1
f
-fo
fo
0
j
-j
f
0
Ir(j2πf)
Qr(j2πf)
f
0
2
2
Transmitter Output
f
-fo fo0
1 1
f
0
f
-fo
fo
0
j
-j
2cos(2πfot)
2sin(2πfot)
y(t)
it(t)
qt(t)
It(j2πf)
Qt(j2πf)
f
0
f
-fo fo0
1 11
1
f
-fo
fo
0
j
-j
Yi(j2πf)
Yq(j2πf)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
I/Q Demodulation
• Demodulate with both a cosine and sine wave
– Both I and Q channels are recovered!
• I/Q modulation allows twice the amount of 
information to be sent compared to basic AM 
modulation with same bandwidth
What can go wrong here?
• Modulate each with a cosine & sine, then sum the result
In-Phase (I) 
component (real)
Quadrature (Q) 
component (imaginary)
• I, Q signals occupy the same frequency band
– One is real (for cos), one is imaginary (for sin)
• Demodulate with both a sine and a cosine
– Both I and Q channels are recovered!
8
In-Phase/Quadrature Demodulation
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 7
Receiver Output
2cos(2πfot)
2sin(2πfot)
y(t)
ir(t)
qr(t)
f
f
f
-fo fo0
1 1
f
-fo
fo
0
j
-j
f
0
Ir(j2πf)
Qr(j2πf)
f
0
2
2
Transmitter Output
f
-fo fo0
1 1
f
0
f
-fo
fo
0
j
-j
2cos(2πfot)
2sin(2πfot)
y(t)
it(t)
qt(t)
It(j2πf)
Qt(j2πf)
f
0
f
-fo fo0
1 11
1
f
-fo
fo
0
j
-j
Yi(j2πf)
Yq(j2πf)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
I/Q Demodulation
• Demodulate with both a cosine and sine wave
– Both I and Q channels are recovered!
• I/Q modulation allows twice the amount of 
information to be sent compared to basic AM 
modulation with same bandwidth
What can go wrong here?
Transmitter: Receiver:
• I and Q channels get swapped at receiver
– Key observation: No information is lost!
9
I/Q Demodulation: 90º Phase Shift
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 8
f
-fo fo
0
-1 -1
f
-fo
fo
0
j
-j
Receiver Output
2sin(2πfot)
-2cos(2πfot)
y(t)
ir(t)
qr(t)
f
0
Ir(j2πf)
Qr(j2πf)
f
0
-2
2
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Transmitter Output
f
-fo fo0
1 1
f
0
f
-fo
fo
0
j
-j
2cos(2πfot)
2sin(2πfot)
y(t)
it(t)
qt(t)
It(j2πf)
Qt(j2πf)
f
0
f
-fo fo0
1 11
1
f
-fo
fo
0
j
-j
Yi(j2πf)
Yq(j2πf)
f
f
Impact of 90 Degree Phase Shift
• I and Q channels get swapped at receiver
– Key observation: no information is lost!
• Questions
– What would happen with a small frequency offset?
– What would happen with a large frequency offset?
Transmitter: Receiver:
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 9
f
0
Receiver Output
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
It(j2πf)
Qt(j2πf)
f
0
1
1
f
0
Ir(j2πf)
Qr(j2πf)
f
0
2
2
2cos(2πfot)
2sin(2πfot)
Baseband Input
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Summary of Analog I/Q Modulation
• Frequency domain view
t
t
t
t
Receiver Output
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Baseband Input
Lowpass
H(j2πf)
Lowpass
H(j2πf)
• Time domain view
10
Summary of Analog I/Q modulation
• I/Q modulation allows twice the amount of information to be 
sent compared to basic AM
• Impact of phase offset is to swap I/Q
• Impact of frequency offset is I/Q swapping (small offset) 
– Or catastrophic corruption (large offset) of received signal 
11
I/Q modulation: Wrap-up
1. Analog I/Q modulation
2. Discrete-time processing of continuous signals
– The Digital Abstraction
– Quantization: Discretizing values
– Sequences: Discretizing time
3. Digital I/Q Modulation
12
Roadmap
• Encoding of pixel at location (x, y) in 
a Black & White picture:
– 0 Volts = Black
– 1 Volt = White
– 0.37 V = 37% Gray
– etc...
• Encoding of a picture:
– Scan points in prescribed order
– Generate a continuous voltage 
waveform (baseband signal)
13
Representing information with voltage
6.082 Spring 2007 The Digital Abstraction, Slide 3
Information Processing in the Analog Domain
First let’s introduce some processing blocks:
vCopyv
INVv 1-v
6.082 Spring 2007 The Digital Abstraction, Slide 3
Information Processing in the Analog Domain
First let’s introduce some processing blocks:
vCopyv
INVv 1-v
6.082 Spring 2007 Digital Modulation (Part I), Slide 4
f
0
Receiver Output
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
It(j2πf)
Qt(j2πf)
f
0
1
1
f
0
Ir(j2πf)
Qr(j2πf)
f
0
2
2
2cos(2πfot)
2sin(2πfot)
Baseband Input
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Summary of Analog I/Q Demodulation
• Frequency domain view
t
t
t
t
Receiver Output
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Baseband Input
Lowpass
H(j2πf)
Lowpass
H(j2πf)
• Time domain view14
Let’s build a P2P television network
6.082 Spring 2007 The Digital Abstraction, Slide 3
Information Processing in the Analog Domain
First let’s introduce some processing blocks:
vCopyv
INVv 1-v
x(t)
∿
×
∿
×
6.082 Spring 2007 Digital Modulation (Part I), Slide 4
f
0
Receiver Output
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
It(j2πf)
Qt(j2πf)
f
0
1
1
f
0
Ir(j2πf)
Qr(j2πf)
f
0
2
2
2cos(2πfot)
2sin(2πfot)
Baseband Input
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Summary of Analog I/Q Demodulation
• Frequency domain view
t
t
t
t
Receiver Output
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Baseband Input
Lowpass
H(j2πf)
Lowpass
H(j2πf)
• Time domain view
6.082 Spring 2007 The Digital Abstraction, Slide 3
Information Processing in the Analog Domain
First let’s introduce some processing blocks:
vCopyv
INVv 1-v
Encode Decode
One hop:
6.082 Spring 2007 The Digital Abstraction, Slide 5
?
Let’s build a system!
Copy INV
Copy INV
Copy INV
Copy INV
output
(In Theory) (Reality) 
input
6.082 Spring 2007 The Digital Abstraction, Slide 3
Information Processing in the Analog Domain
First let’s introduce some processing blocks:
vCopyv
INVv 1-v
Many hops:
DecodeEncode
1. Transmitter doesn’t work right
2. Receiver doesn’t work right
3. Theory is imperfect
4. System architecture isn’t right
• Answer: All of the above!
• Noise, transmitter/receiver imperfections inevitable, so must 
design system to tolerate some amount of error
15
Why did our network fail?
• Problem: It’s hard to distinguish legitimate analog 
waveforms from corrupted ones
– Every waveform is potentially legitimate!
• Small errors accumulate at each hop
• Endpoints can’t help, so need to eliminate errors at each hop
16
Analog communications issues
6.082 Spring 2007 The Digital Abstraction, Slide 16
Plan: Mixed Signal Architecture
Analog
to
Digital
store
xmit/rcv
modify
Digital
to
Analog
ANALOG ANALOGDIGITAL
Volts
Freq
Phase
Volts
Freq
Phasebitsbits
Digital
Signal
Processing
Continuous time,
Continuous values
Discrete time,
Discrete values
17
Plan: Mixed Signal Architecture
• Continuous-time, continuous-valued waveform is being 
converted into a discrete-valued sequence
– Only certain values are ”allowed” to be used
• Questions yet to be answered:
1. How many discrete values should we use?
2. How rapidly in time should we sample the waveform?
18
Discrete time, Discrete values
1. Analog I/Q modulation
2. Discrete-time processing of continuous signals
– The Digital Abstraction
– Quantization: Discretizing values
– Sequences: Discretizing time
3. Digital I/Q Modulation
19
Roadmap
• First encoding attempt:
• But, what happens in the presence of noise?
20
Encoding Information Digitally
6.082 Spring 2007 The Digital Abstraction, Slide 10
Forbidden Zone
Digital Signaling II
volts
0 VDDVL
0 1
Encoding Attempt #2:
VH
This avoids “close calls”, but now we have to consider noise (i.e.
unavoidable perturbations to our signaling voltage)
DEVICE
#2
DEVICE
#1
NOISE
V V±N
So an output voltage just
below VL might become an
illegal input voltage in the
forbidden zone!
6.082 Spring 2007 The Digital Abstraction, Slide 10
Forbidden Zone
Digital Signaling II
volts
0 VDDVL
0 1
Encoding Attempt #2:
VH
This avoids “close calls”, but now we have to consider noise (i.e.
unavoidable perturbations to our signaling voltage)
DEVICE
#2
DEVICE
#1
NOISE
V V±N
So an output voltage just
below VL might become an
illegal input voltage in the
forbidden zone!
6.082 Spring 2007 The Digital Abstraction, Slide 11
Big Idea: Noise Margins
OUTPUTS:
INPUTS:
Forbidden Zone
volts
0 VDDVOL
0OUT 1OUT
VOH
volts
0 VDDVIL
0IN 1IN
VIHVOL VOH
Noise Margins
Let’s leave room for bad things to happen! So we’ll design
devices restore marginally valid input signals. They must
accept marginal inputs and provide unquestionable outputs
(i.e., to leave room for noise).
21
Big Idea: Noise Margins
1. Analog I/Q modulation
2. Discrete-time processing of continuous signals
– The Digital Abstraction
– Quantization: Discretizing values
– Sequences: Discretizing time
3. Digital I/Q Modulation
22
Roadmap
23
Quantization: How many discrete values?
6.082 Spring 2007 The Digital Abstraction, Slide 18
Discrete Values
If we use N bits to encode the magnitude of one of the
discrete-time samples, we can capture 2N possible values.
So we’ll divide up the range of possible sample values into
2N intervals and choose the index of the enclosing interval
as the encoding for the sample value.
sample voltage
quantized value 1
1-bit
11
2-bit
110
3-bit
1101
4-bit
0
0
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
1
2
3
1
VMAX
VMIN
A
na
lo
g 
va
lu
e
 c
on
ve
rt
e
d
 t
o 
d
ig
it
a
l 
in
d
e
x
• Introduce at most ½ interval length of quantization error
24
Quantization Error
6.082 Spring 2007 The Digital Abstraction, Slide 19
Quantization Error
53
54
55
56
57
Note that when we quantize the scaled sample values we
may be off by up to ±½ step from the true sampled values.
54 55 56 55 55
The red shaded region shows
the error we’ve introduced
1. Analog I/Q modulation
2. Discrete-time processing of continuous signals
– The Digital Abstraction
– Quantization: Discretizing values
– Sequences: Discretizing time
3. Digital I/Q Modulation
25
Roadmap
• The boundary between analog and digital
– Real world filled with continuous-time signals
– Computers operate on discrete-time sequences
• Crossing analog to digital boundary requires conversion between the two
26
Need for Continuous-to-Discrete Conversion
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 2
The Need for Sampling
• The boundary between analog and digital
– Real world is filled with continuous-time signals
– Computers (i.e. Matlab) operate on sequences
• Crossing the analog-to-digital boundary requires 
sampling of the continuous-time signals
• Key questions
– How do we analyze the sampling process?
– What can go wrong?
x[n]xc(t)
t
xc(t)
n
x[n]
1
A-to-D
Converter
Real World
(USRP Board)
Matlab
Real world (Hack RF) Computer (Python)
Continuous-time signal Discrete-time sequence
• Too slow (undersampling):
– DT sequence loses information about CT signal
• Too fast (oversampling):
– DT sequence contains redundant, useless information
27
How Fast to Sample?
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 2
The Need for Sampling
• The boundary between analog and digital
– Real world is filled with continuous-time signals
– Computers (i.e. Matlab) operate on sequences
• Crossing the analog-to-digital boundary requires 
sampling of the continuous-time signals
• Key questions
– How do we analyze the sampling process?
– What can go wrong?
x[n]xc(t)
t
xc(t)
n
x[n]
1
A-to-D
Converter
Real World
(USRP Board)
Matlab
Real world (Hack RF) Computer (Python)
Continuous-time (CT) signal Discrete-time (DT) sequence
• Two-step process:
1. Sample continuous-time signal every T seconds
– Model as multiplication of signal with impulse train
2. Create sequence from amplitude of scaled impulses
– Model as rescaling of time axis (T à 1)
28
Modelling Continuous-to-Discrete Conversion
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 3
An Analytical Model for Sampling
• Two step process
– Sample continuous-time signal every T seconds
• Model as multiplication of signal with impulse train
– Create sequence from amplitude of scaled impulses
• Model as rescaling of time axis (T → 1)
• Notation: replace impulses with stem symbols
p(t)
xp(t) x[n]xc(t) Impulse Train
to Sequence
t
p(t)
T
1
t
T
t
xc(t) xp(t)
n
x[n]
1
Can we model this in the frequency domain?
• Impulse train in time corresponds to impulse train in frequency
– Spacing in time of T seconds corresponds to 1/T Hz spacing in frequency
29
Fourier Transform of Impulse Train
6.082Spring 2007 I/Q Modulation and RC Filtering, Slide 4
Fourier Transform of Impulse Train
• Impulse train in time corresponds to impulse train 
in frequency
– Spacing in time of T seconds corresponds to spacing in 
frequency of 1/T Hz
– Scale factor of 1/T for impulses in frequency domain
– Note: this is painful to derive, so we won’t …
• The above transform pair allows us to see the 
following with pictures
– Sampling operation in frequency domain
– Intuitive comparison of FT, DTFT, and Fourier Series
t
p(t)
T
1
P(j2 πf)
t
0
T
1
T
2
T
-1
T
-2
T
1
• Sampling in time leads to a periodic Fourier Transform, with period 1/T 
30
Frequency Domain view of Sampling
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 5
Frequency Domain View of Sampling
• Recall that multiplication in time corresponds to 
convolution in frequency
• We see that sampling in time leads to a periodic
Fourier Transform with period 1/T
t
p(t)
T
1
P(j2 πf)
t
0
T
1
T
2
T
-1
T
-2
T
1
t
T
t
xc(t) xp(t)
Xp(j2 πf)
f
0
T
1
T
2
T
-1
T
-2
T
A
Xc(j2 πf)
f
A
• Conversion to a sequence amounts to setting T = 1
• Resulting Fourier Transform is now periodic with a period of one
31
Frequency Domain View of a Sequence
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 6
t
T
xp (t)
n
x[n]
1
Xp (j2 πf)
f
0
T
1
T
2
T
-1
T
-2
T
A
X(e j2 πλ)
λ
0 1 2-1-2
T
A
Frequency Domain View of Output Sequence
• Scaling in time leads to scaling in frequency
– Compression/expansion in time leads to expansion/ 
compression in frequency
• Conversion to sequence amounts to T→ 1
– Resulting Fourier Transform is now periodic with period 1
– Note that we are now essentially dealing with the DTFT
• Sampling leads to periodicity in frequency domain
32
Summary: Continuous-to-Discrete Conversion
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 7
p(t)
xp(t) x[n]xc(t) Impulse Train
to Sequence
t
p(t)
T
1
t
T
t
xc(t) xp(t)
n
x[n]
1
Xp(j2 πf)
f
0
T
1
T
2
T
-1
T
-2
T
A
X(ej2 πλ)
λ
0 1 2-1-2
T
A
Xc(j2 πf)
f
0
A
Summary of Sampling Process
• Sampling leads to periodicity in frequency domain
We need to avoid overlap of replicated 
signals in frequency domain (i.e., aliasing)
• No overlap in frequency domain (aliasing) when 
⁄"# $ − &'( ≥ &'( or ⁄# $ ≥ 2&'(
• We refer to the minimum 1/T that avoids aliasing as the Nyquist sampling 
frequency of a signal
33
The Sampling Theorem
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 8
t
p(t)
T
1
P(j2 πf)
t
0
T
1
T
2
T
-1
T
-2
T
1
t
T
t
xc(t) xp(t)
Xp(j2 πf)
f
0
T
1
T
2
T
-1
T
-2
T
A
Xc(j2 πf)
f
A
fbw-fbw
fbw-fbw
- fbw
T
1- + fbw
T
1
The Sampling Theorem
• Overlap in frequency domain (i.e., aliasing) is 
avoided if:
• We refer to the minimum 1/T that avoids aliasing 
as the Nyquist sampling frequency
• Time domain: Resulting sequence maintains same period as input signal
34
Sine Wave Example:
Sampling Above Nyquist Rate
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 9
p(t)
xp(t)
T
Impulse Train
to Sequence
x[n]
T 1
xc(t)
tt n
t
1/T0-1/T
f
10-1
λ
1/T0-1/T
f
Example: Sample a Sine Wave
• Time domain: resulting sequence maintains the 
same period as the input continuous-time signal
• Frequency domain: no aliasing
Xc(j2πf) Xp(j2πf) X(j2πλ)
Sample rate is well above Nyquist rate
• Time Domain: Resulting sequence maintains same period as input signal
35
Sine Wave Example:
Sampling at Nyquist Rate
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 10
tt n
p(t)
xp(t)
T
Impulse Train
to Sequence
x[n]
T 1
xc(t)
t
1/T0-1/T
f
10-1
λ
1/T0-1/T
f
Increase Input Frequency Further …
Xc(j2πf) Xp(j2πf) X(j2πλ)
• Time domain: resulting sequence still maintains the 
same period as the input continuous-time signal
• Frequency domain: no aliasing
Sample rate is at Nyquist rate
• Sequence now appears as constant (zero frequency) signal
• Frequency domain: Aliasing to 0
36
Sine Wave Example:
Sampling at Half the Nyquist Rate
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 11
tt n
p(t)
xp(t)
T
Impulse Train
to Sequence
x[n]
T 1
xc(t)
t
1/T0-1/T
f
10-1
λ
1/T0-1/T
f
Increase Input Frequency Further …
Xc(j2πf) Xp(j2πf) X(j2πλ)
• Time domain: resulting sequence now appears as a 
DC signal!
• Frequency domain: aliasing to DC
Sample rate is at half the Nyquist rate
• Resulting sequence now a sine wave with a different period than the input
• Frequency domain: Aliasing to a lower frequency
37
Sine Wave Example:
Sampling Below the Nyquist Rate
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 12
tt n
p(t)
xp(t)
T
Impulse Train
to Sequence
x[n]
T 1
xc(t)
t
1/T0-1/T
f
10-1
λ
1/T0-1/T
f
Increase Input Frequency Further …
Xc(j2πf) Xp(j2πf) X(j2πλ)
• Time domain: resulting sequence is now a sine 
wave with a different period than the input
• Frequency domain: aliasing to lower frequency
Sample rate is well below the Nyquist rate
• Typically set sample rate to exceed desired signal’s bandwidth
• Real systems can introduce noise/other interference at high frequencies
– Sampling causes noise to alias into desired frequency band
38
The Issue of High Frequency Noise
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 13
The Issue of High Frequency Noise
• We typically set the sample rate to be large 
enough to accommodate full bandwidth of signal
• Real systems often introduce noise or other 
interfering signals at higher frequencies
– Sampling causes this noise to alias into the desired 
signal band
p(t)
xp(t) x[n]xc(t) Impulse Train
to Sequence
t
p(t)
T
1
Xp(j2 πf)
f
0
T
1
T
2
T
-1
T
-2
T
A
X(ej2 πλ)
λ
0 1 2-1-2
T
A
Xc(j2 πf)
f
0
A
T
1
T
-1
Undesired
Signal or
Noise
• Practical systems use a continuous-time filter before the sampling operation
– Removes noise/interference >1/2T in frequency, prevents aliasing
39
Anti-Alias Filtering
6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 14
xc(t)
H(j2 πf)
p(t)
xp(t) x[n]Impulse Train
to Sequence
t
p(t)
T
1
Xp(j2 πf)
f
0
T
1
T
2
T
-1
T
-2
X(ej2 πλ)
λ
0 1 2-1-2
Xc(j2 πf)
f
0
T
1
T
-1
xc(t)
Anti-alias
Lowpass
H(j2 πf)
Anti-Alias Filtering
• Practical A-to-D converters include a continuous-
time filter before the sampling operation
– Designed to filter out all noise and interfering signals 
above 1/(2T) in frequency
– Prevents aliasing
• Allows error correction to be achieved
– Less sensitivity to radio channel imperfections
• Enables compression of information
– More efficient use of the channel
• Supports a wide variety of information content
– Voice, text, video can all be represented as digital bit streams
40
Summary: Advantages of Going Digital
1. Analog I/Q modulation
2. Discrete-time processing of continuous signals
3. Digital I/Q Modulation
41
Roadmap
• Leverage analog communication channel to send discrete-valued symbols
– Example: Send symbol from set { -3, -1, 1, 3 } on both I and Q channels 
• At receiver, sample I/Q waveforms every symbol period
– Associate each sampled I/Q value with symbols from same set on both I 
and Q channels
42
Digital I/Q modulation
6.082 Spring 2007 Digital Modulation (Part I), Slide 5
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(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
I
Q
6.082 Spring 2007 Digital Modulation (Part I), Slide 7
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
LowpassH(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
-3 -1 1 3
-3
-1
1
3
I
Q
I
Q
Constellation Diagrams
• Plot I/Q samples on x-y axis
– Example: sampled I/Q value of 
{1,-3} forms a dot at x=1, y=-3
– As more samples are plotted, 
constellation diagram eventually 
displays all possible symbol values
• Constellation diagram provides 
a sense of how easy it is to 
distinguish between different 
symbols 
• Plot I/Q samples on x-y axis
– As samples are plotted, constellation diagram 
eventually displays all possible symbol values
• Provides a sense of how easy it is to distinguish
between different symbols
43
Constellation Diagrams
6.082 Spring 2007 Digital Modulation (Part I), Slide 7
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
-3 -1 1 3
-3
-1
1
3
I
Q
I
Q
Constellation Diagrams
• Plot I/Q samples on x-y axis
– Example: sampled I/Q value of 
{1,-3} forms a dot at x=1, y=-3
– As more samples are plotted, 
constellation diagram eventually 
displays all possible symbol values
• Constellation diagram provides 
a sense of how easy it is to 
distinguish between different 
symbols 
6.082 Spring 2007 Digital Modulation (Part I), Slide 8
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
00 01 11 10
00
01
11
10
I
Q
I
Q
Sending Digital Bits
• Assign each I/Q symbol to a 
set of digital bits
– Example: I/Q = {1,3} translates 
to bits of 1110
– Gray coding minimizes bit errors
when symbol errors are made
• Example: I/Q = {1,1} translates 
to bits of 1010
– Only one bit change from 
I/Q = {1,3}
• Assign bits to each I/Q symbol
– Example: I/Q = {1, 3} translates to bits 1110
– Gray coding minimizes bit errors when symbol 
errors are made
• Example: I/Q = {3, 3} translates to bits 1010 
(one bit flip from {1, 3})
44
Sending Digital Bits
6.082 Spring 2007 Digital Modulation (Part I), Slide 8
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
00 01 11 10
00
01
11
10
I
Q
I
Q
Sending Digital Bits
• Assign each I/Q symbol to a 
set of digital bits
– Example: I/Q = {1,3} translates 
to bits of 1110
– Gray coding minimizes bit errors
when symbol errors are made
• Example: I/Q = {1,1} translates 
to bits of 1010
– Only one bit change from 
I/Q = {1,3}
6.082 Spring 2007 Digital Modulation (Part I), Slide 9
00 01 11 10
00
01
11
10
I
Q
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
Receiver
Noise I
Q
The Impact of Noise
• Noise perturbs sampled I/Q 
values
– Constellation points no longer 
consist of single dots for each 
symbol
• Issue: what is the best 
way to match received I/Q 
samples with their 
corresponding symbols?
• At receiver, noise perturbs sampled I/Q values
– Constellation points no longer consist of single 
points for each symbol
• Issue: What’s the best way of matching received 
I/Q samples with their corresponding transmitted 
symbols?
45
The Impact of Noise
6.082 Spring 2007 Digital Modulation (Part I), Slide 9
00 01 11 10
00
01
11
10
I
Q
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
Receiver
Noise I
Q
The Impact of Noise
• Noise perturbs sampled I/Q 
values
– Constellation points no longer 
consist of single dots for each 
symbol
• Issue: what is the best 
way to match received I/Q 
samples with their 
corresponding symbols?
6.082 Spring 2007 Digital Modulation (Part I), Slide 10
Decision
Boundaries I
Q
Decision
Boundaries
00 01 11 10
00
01
11
10
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
Receiver
Noise I
Q
Symbol Selection Based on Slicing
• Match I/Q samples to 
their corresponding 
symbols based on 
decision regions
– Choose decision regions 
to minimize symbol 
errors
– Decision boundaries are 
also called slicing levels
• Receiver matches I/Q samples to corresponding symbols 
based on decision boundaries
• Decision boundaries are also called slicing levels
46
Symbol Selection based on Slicing
6.082 Spring 2007 Digital Modulation (Part I), Slide 10
Decision
Boundaries I
Q
Decision
Boundaries
00 01 11 10
00
01
11
10
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
Receiver
Noise I
Q
Symbol Selection Based on Slicing
• Match I/Q samples to 
their corresponding 
symbols based on 
decision regions
– Choose decision regions 
to minimize symbol 
errors
– Decision boundaries are 
also called slicing levels
6.082 Spring 2007 Digital Modulation (Part I), Slide 11
I
Q
00 01 11 10
00
01
11
10
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
Receiver
Noise I
Q
Transitioning Between Symbols
• Transition behavior between 
symbols is influenced by both 
transmit I/Q input waveforms 
and receive filter
– We will focus on impact of 
transition behavior at 
transmitter today
– Ignore the impact of noise for 
this analysis
• Transition behavior influenced by transmit I/Q input 
waveforms and receive filter
– Today: Focus on what the transmitter does
– Ignore impact of noise for this analysis
47
Transitioning Between Symbols
6.082 Spring 2007 Digital Modulation (Part I), Slide 11
I
Q
00 01 11 10
00
01
11
10
2cos(2πfot)
2sin(2πfot)
ir(t)
qr(t)
it(t)
qt(t)
2cos(2πfot)
2sin(2πfot)
Lowpass
H(j2πf)
Lowpass
H(j2πf)
Receiver Output
t
t
t
Baseband Input
t
Sample
Times
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
3
1
-1
-3
Receiver
Noise I
Q
Transitioning Between Symbols
• Transition behavior between 
symbols is influenced by both 
transmit I/Q input waveforms 
and receive filter
– We will focus on impact of 
transition behavior at 
transmitter today
– Ignore the impact of noise for 
this analysis
48
Transitions and the Transmitted Spectrum
• Want transmitted spectrum with minimal 
bandwidth
• Issue: Sharply changing I/Q leads to very 
wide bandwidth spectrum
49
Impact of Transmit Filter
• Transmit filter shapes the pulses, 
enables reduced bandwidth for 
transmitted spectrum
• Next Issue: Can lead to bit errors
• Lowering the transmit bandwidth 
leads to increased bit errors
50
Impact of Low Bandwidth Transmit Filter
• Eye Diagram: Wrap signal back onto itself in periodic time intervals,
retaining all “traces”
51
Eye Diagrams
• Increasing the number of symbols eventually reveals all possible symbol 
transition trajectories
– Intuitively displays the impact of filtering on transmitted signal 
52
Looking at Many Symbols
53
Assessing the Quality of an Eye Diagram
• Eye diagram allows visual inspection of the impact of sample time and 
decision boundary choices
– Large “eye opening” implies less vulnerability to symbol errors
Sample times:
• Open eye diagrams leadto tight 
symbol groupings in constellation
– Assumes proper sample time 
placement
54
Relating Eye Diagrams to Constellation
• Eye diagrams show increased 
inter-symbol interference (ISI)
– Also show symbol decision 
sensitivity to sample timing 
placement
55
Impact of Low Transmit Bandwidth
• Digital modulation: Sends discrete-valued symbols through an analog
communication channel 
– Receiver must sample I/Q signals at the appropriate time 
– Receiver matches I/Q sample values to corresponding symbols based on 
decision regions 
– Constellation diagrams are a convenient tool to see likelihood of bit 
errors being made 
• Choice of transmit filter: Tradeoff between achieving a minimal bandwidth
transmitted spectrum and minimal ISI 
– Eye diagrams: A convenient tool to see effects of ISI and sensitivity of bit 
errors to sample time choice 
56
Digital Modulation: Summary
Thursday Topic:
Digital Communications II:
Receiver Processing, Performance
57