6.003 lecture 23: quantization and modulation
TRANSCRIPT
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6.003: Signals and Systems
Quantization and Modulation
May 4, 2010
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Last Time
Quantization: discrete representation of amplitudes.
1
0
2 bits 4 bits
00
01
10 3 bits
Out
put v
olta
ge
-1 -1 0 1 -1 0 1 -1 0 1
Input voltage Input voltage Input voltage
0 0.5 1 -1
0
1
Time (second) 0 0.5 1
Time (second) 0 0.5 1
Time (second)
Digital representations of signals require discrete representations of
both the independent variable (time, space, ...) and dependent
variable (amplitude).
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Quantizing Images
Quantizing creates distortion.
8 bits 6 bits 4 bits
3 bits 2 bits 1 bit
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Quantization
Adding noise (dither) can reduce the perceptual effect of quantiza
tion.
Quantization: y = Q(x)
Quantization with dither: y = Q(x + n)
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Quantizing Images
Adding noise (dither) reduces “banding” in images.
3 bits 2 bits 1 bit
3 bits dither 2 bits dither 1 bit dither
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Quantizing Sound
Adding noise (dither) can reduces “distortions” in sound.
• 16 bits/sample
• 4 bits/sample
• 4 bits with dither/sample
• 3 bits/sample
• 3 bits with dither/sample
• 2 bit/sample
• 2 bit with dither/sample
J.S. Bach, Sonata No. 1 in G minor Mvmt. IV. Presto
Nathan Milstein, violin
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Robert’s Technique
Robert’s technique decorrelates quantization errors from the signal.
Quantization: y = Q(x)
Quantization with dither: y = Q(x + n)
Quantization with Robert’s technique: y = Q(x + n) − n
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Quantizing Images
Robert’s technique decorrelates quantization errors from the signal.
3 bits dither 2 bits dither 1 bit dither
3 bits Robert’s 2 bits Robert’s 1 bit Roberts
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Quantizing Sound
Robert’s technique decorrelates quantization errors from the signal.
• 16 bits/sample
• 2 bits/sample
• 2 bits with dither/sample
• 2 bits with Robert’s technique/sample
J.S. Bach, Sonata No. 1 in G minor Mvmt. IV. Presto
Nathan Milstein, violin
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Fourier Analysis
Quantization
x(t) |X(jω)|
t
ω
Dither
x(t) |X(jω)|
t
ω
Robert’s Technique
x(t) |X(jω)|
t
ω
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Modulation
Applications of signals and systems in communication systems.
Example: Transmit voice via telephone wires (copper)
mic amp telephone wire amp speaker
Works well: basis of local land-based telephones.
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Wireless Communication
In cellular communication systems, signals are transmitted via elec
tromagnetic (E/M) waves.
mic amp E/M wave amp speaker
For efficient transmission and reception, antenna length should be
on the order of the wavelength.
Telephone-quality speech contains frequencies from 200 to 3000 Hz.
How long should the antenna be?
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Check Yourself
For efficient transmission and reception, the antenna length
should be on the order of the wavelength.
Telephone-quality speech contains frequencies between 200 Hz
and 3000 Hz.
How long should the antenna be?
1. < 1 mm
2. ∼ cm
3. ∼ m
4. ∼ km
5. > 100 km
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Check Yourself
Wavelength is λ = c/f so the lowest frequencies (200 Hz) produce
the longest wavelengths
λ = c = 3 × 108 m/s = 1.5 × 106 m = 1500 km . f 200 Hz
and the highest frequencies (3000 Hz) produce the shortest wave
lengths
λ = c = 3 × 108 m/s = 105 m = 100 km . f 3000 Hz
On the order of hundreds of miles!
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Check Yourself
For efficient transmission and reception, the antenna length
should be on the order of the wavelength.
Telephone-quality speech contains frequencies between 200 Hz
and 3000 Hz.
How long should the antenna be? 5
1. < 1 mm
2. ∼ cm
3. ∼ m
4. ∼ km
5. > 100 km
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Check Yourself
What frequency E/M wave is well matched to an antenna
with a length of 10 cm (about 4 inches)?
1. < 100 kHz
2. 1 MHz
3. 10 MHz
4. 100 MHz
5. > 1 GHz
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Check Yourself
A wavelength of 10 cm corresponds to a frequency of
3 × 108 m/sf = c ∼ ≈ 3 GHz . λ 10 cm
Modern cell phones use frequencies near 2 GHz.
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Check Yourself
What frequency E/M wave is well matched to an antenna
with a length of 10 cm (about 4 inches)? 5
1. < 100 kHz
2. 1 MHz
3. 10 MHz
4. 100 MHz
5. > 1 GHz
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Wireless Communication
Speech is not well matched to the wireless medium.
Many applications require the use of signals that are not well
matched to the required media.
signal applications
audio telephone, radio, phonograph, CD, cell phone, MP3
video television, cinema, HDTV, DVD
internet coax, twisted pair, cable TV, DSL, optical fiber, E/M
We can often modify the signals to obtain a better match.
Today we will introduce simple matching strategies based on
modulation.
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Check Yourself
Construct a signal Y that codes the audio frequency information
in X using frequency components near 2 GHz.
ω
|X(jω)|
ω ωc
|Y (jω)|
Determine an expression for Y in terms of X.
1. y(t) = x(t) e jωct 2. y(t) = x(t) ∗ e jωct 3. y(t) = x(t) cos(ωct) 4. y(t) = x(t) ∗ cos(ωct)
5. none of the above
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Check Yourself
Construct a signal Y that codes the audio frequency information
in X using frequency components near 2 GHz.
ω
|X(jω)|
ω ωc
|Y (jω)|
Determine an expression for Y in terms of X. 1
1. y(t) = x(t) e jωct 2. y(t) = x(t) ∗ e jωct 3. y(t) = x(t) cos(ωct) 4. y(t) = x(t) ∗ cos(ωct)
5. none of the above
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Amplitude Modulation
Multiplying a signal by a sinusoidal carrier signal is called amplitude
modulation (AM). AM shifts the frequency components of X by ±ωc.×x(t) y(t)
cos ωct
|X(jω)|
ω
ω−ωc ωc
|Y (jω)|
ω −ωc ωc
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Amplitude Modulation
Multiplying a signal by a sinusoidal carrier signal is called amplitude
modulation. The signal “modulates” the amplitude of the carrier.
x(t) × y(t)
cos ωct
x(t) t
cos ωct
t
t
x(t) cos ωct
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Amplitude Modulation
How could you recover x(t) from y(t)?
×x(t) y(t)
cos ωct
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Synchronous Demodulation
X can be recovered by multiplying by the carrier and then low-pass
filtering. This process is called synchronous demodulation.
y(t) = x(t) cos ωct
1 1 z(t) = y(t) cos ωct = x(t) × cos ωct × cos ωct = x(t) + cos(2ωct)2 2
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Synchronous Demodulation
Synchronous demodulation: convolution in frequency.
|Y (jω)|
ω −ωc ωc
ω−ωc ωc
|Z(jω)|
ω −2ωc 2ωc
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Synchronous Demodulation
We can recover X by low-pass filtering.
|Y (jω)|
ω −ωc ωc
ω−ωc ωc
|Z(jω)|
ω
2
−2ωc 2ωc
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Frequency-Division Multiplexing
Multiple transmitters can co-exist, as long as the frequencies that
they transmit do not overlap.
x1(t)z1(t)
cos 1t
z2(t) z(t)
z3(t)
cos 2t cos ct
cos 3t
LPFx2(t) y(t)
x3(t)
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Frequency-Division Multiplexing
Multiple transmitters simply sum (to first order).
z1(t) x1(t)
cos 1t
x2(t) z2(t) z(t)
z3(t)
cos 2t cos ct
cos 3t
LPF y(t)
x3(t)
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Frequency-Division Multiplexing
The receiver can select the transmitter of interest by choosing the
corresponding demodulation frequency.
Z(j )
X1(j )
1 2 3
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Frequency-Division Multiplexing
The receiver can select the transmitter of interest by choosing the
corresponding demodulation frequency.
Z(j )
X2(j )
1 2 3
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Frequency-Division Multiplexing
The receiver can select the transmitter of interest by choosing the
corresponding demodulation frequency.
Z(j )
X3(j )
1 2 3
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Broadcast Radio
“Broadcast” radio was championed by David Sarnoff, who previously
worked at Marconi Wireless Telegraphy Company (point-to-point).
• envisioned “radio music boxes”
• analogous to newspaper, but at speed of light
• receiver must be cheap (as with newsprint)
• transmitter can be expensive (as with printing press)
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Inexpensive Radio Receiver
The problem with making an inexpensive radio receiver is that you
must know the carrier signal exactly!
x(t) y(t)z(t) LPF
cos( ct) cos( ct+ )
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Check Yourself
The problem with making an inexpensive radio receiver is that
you must know the carrier signal exactly!
z(t)x(t) y(t)
cos( ct) cos( ct+ )
LPF
What happens if there is a phase shift φ between the signal
used to modulate and that used to demodulate?
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Check Yourself
y(t) = x(t) × cos(ωct) × cos(ωct + φ)
1 1 = x(t) × 2 cos φ + 2
cos(2ωct + φ)
Passing y(t) through a low pass filter yields 12 x(t) cos φ.
If φ = π/2, the output is zero!
If φ changes with time, then the signal “fades.”
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AM with Carrier
One way to synchronize the sender and receiver is to send the carrier
along with the message.
× +
C
x(t) z(t)
cos ωct
z(t) = x(t) cos ωct + C cos ωct = (x(t) + C) cos ωct
t
z(t)
x(t) + C
Adding carrier is equivalent to shifting the DC value of x(t).If we shift the DC value sufficiently, the message is easy to decode:
it is just the envelope (minus the DC shift).
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Inexpensive Radio Receiver
If the carrier frequency is much greater than the highest frequency
in the message, AM with carrier can be demodulated with a peak
detector.
z(t) z(t)
y(t)
R C ty(t)
In AM radio, the highest frequency in the message is 5 kHz and the
carrier frequency is between 500 kHz and 1500 kHz.
This circuit is simple and inexpensive.
But there is a problem.
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Inexpensive Radio Receiver
AM with carrier requires more power to transmit the carrier than to
transmit the message!x(t)
xp
xrms
xp > 35xrms
t
Speech sounds have high crest factors (peak value divided by rms
value). The DC offset C must be larger than xp for simple envelope
detection to work.
The power needed to transmit the carrier can be 352 ≈ 1000× that
needed to transmit the message.
Okay for broadcast radio (WBZ: 50 kwatts).
Not for point-to-point (cell phone batteries wouldn’t last long!).
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Inexpensive Radio Receiver
Envelope detection also cannot separate multiple senders.
y(t)
z(t)
tz(t) R C y(t)
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Superheterodyne Receiver
Edwin Howard Armstrong invented the superheterodyne receiver,
which made broadcast AM practical.
Edwin Howard Armstrong also invented and
patented the “regenerative” (positive feedback)
circuit for amplifying radio signals (while he was
a junior at Columbia University). He also invented wide-band FM.
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