1B Paper 6: Communications Handout 2: Analogue Modulation Ramji Venkataramanan Signal Processing and Communications Lab Department of Engineering [email protected]Lent Term 2016 1 / 32 Modulation Modulation is the process by which some characteristic of a carrier wave is varied in accordance with an information bearing signal A commonly used carrier is a sinusoidal wave, e.g., cos(2π f c t ). f c is called the carrier frequency. • We are allotted a certain bandwidth centred around f c for our information signal • E.g. BBC Cambridgeshire: f c = 96 MHz, information bandwidth ≈ 200 KHz • Q: Why is f c usually large? A: Antenna size ∝ λ c ⇒ larger frequency, smaller antennas! 2 / 32
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1B Paper 6: CommunicationsHandout 2: Analogue Modulation
Ramji Venkataramanan
Signal Processing and Communications LabDepartment of [email protected]
Lent Term 2016
1 / 32
Modulation
Modulation is the process by which some characteristic of a carrierwave is varied in accordance with an information bearing signal
A commonly used carrier is a sinusoidal wave, e.g., cos(2πfct).fc is called the carrier frequency.
• We are allotted a certain bandwidth centred around fc for ourinformation signal
• E.g. BBC Cambridgeshire: fc = 96 MHz, informationbandwidth ≈ 200 KHz
Analogue vs. Digital ModulationAnalogue Modulation: A continuous information signal x(t)(e.g., speech, audio) is used to directly modulate the carrier wave.
We’ll study two kinds of analogue modulation:
1. Amplitude Modulation (AM) : Information x(t) modulates theamplitude of the carrier wave
2. Frequency Modulation (FM): Information x(t) modulates thefrequency of the carrier wave
We’ll learn about:
– Power & bandwidth of AM & FM signals
– Tx & Rx design
In the last 4 lectures, we will study digital modulation:
• x(t) is first digitised into bits
• Digital modulation then used to transport bits across thechannel
3 / 32
Amplitude Modulation (AM)
• Information signal x(t), carrier cos(2πfct)
• The transmitted AM signal is
sAM(t) = [a0 + x(t)] cos(2πfct)
• a0 is a positive constant chosen so that maxt |x(t)| < a0
• The modulation index of the AM signal is defined as
mA =maxt |x(t)|
a0
“The percentage that the carrier’s amplitude varies above andbelow its unmodulated level”
Why is the modulation index important ?ma < 1 is desirable because we can extract the information signalx(t) from the modulated signal by envelope detection.
4 / 32
When modulation index > 1:
• Phase reversals occur
• x(t) cannot be detected bytracing the +ve envelope
5 / 32
AM Receiver - Envelope Detector
sAM(t) Vout(t)RLC
• On the positive half-cycle of the input signal, capacitor Ccharges rapidly up to the peak value of input sAM(t)
• When input signal falls below this peak, diode becomesreverse-biased: capacitor discharges slowly through loadresistor RL
• In the next positive half-cycle, when input signal becomesgreater than voltage across the capacitor, diode conductsagain until next peak value
• Process repeats . . .
Very inexpensive receiver, but envelope detection needs mA < 1.
6 / 32
Circuit Diagram
AM wave input Envelope detector output
7 / 32
Spectrum of AM
Next, let’s look at the spectrum of sAM(t) = [a0 + x(t)] cos(2πfct)
SAM(f ) = F [sAM(t)]
= F[[a0 + x(t)]
(e j2πfc t + e−j2πfc t)
2
]
=a0
2[δ(f − fc) + δ(f + fc)]
︸ ︷︷ ︸carrier
+1
2[X (f − fc) + X (f + fc)]
︸ ︷︷ ︸information
(F [.] denotes the Fourier transform operation)
8 / 32
Example
SAM(f ) =a0
2[δ(f − fc) + δ(f + fc)] +
1
2[X (f − fc) + X (f + fc)]
−fc fc −W
SAM(f)
f
fc +W−fc +W−fc −W fc
0 W−W
X(f)
C
C/2 C/2
a0/2a0/2
9 / 32
Properties of AM
sAM(t) = [a0 + x(t)] cos(2πfct)
SAM(f ) =a0
2[δ(f − fc) + δ(f + fc)] +
1
2[X (f − fc) + X (f + fc)]
1. Bandwidth: From spectrum calculation, we see that if x(t) isa baseband signal with (one-sided) bandwidth W , the AMsignal sAM(t) is passband with bandwidth
BAM = 2W
2. Power: We now prove that the power of the AM signal is
PAM =a20
2+
PX
2
where PX is the power of x(t)
10 / 32
Power of AM signal
PAM = limT→∞
1
T
∫ T
0[a0 + x(t)]2 cos2(2πfct) dt
= limT→∞
1
T
∫ T
0[a0 + x(t)]2
1 + cos(4πfct)
2dt
=a20
2+
PX
2+ lim
T→∞1
T
∫ T
0
[a0 + x(t)]2
2cos(4πfct) dt
We now show that the last the last term is 0.• cos(4πfct) is a high-frequency sinusoid with period Tc = 1
2fc.
• g(t) = (a0+x(t))2
2 is a baseband signal which changes muchmore slowly than cos(4πfct). Hence, with T = nTc , we have
1
T
∫ T
0
g(t) cos(4πfct) dt ≈ 1
nTc
( ∫ Tc
0
g(0) cos(4πfct) dt +
+
∫ 2Tc
Tc
g(Tc) cos(4πfct) dt . . . +
∫ nTc
(n−1)Tc
g((n − 1)Tc) cos(4πfct) dt)
= 0
Hence PAM =a202 + PX
2 .11 / 32
Double Sideband Suppressed Carrier (DSB-SC)The power of AM signal is
PAM =a20
2︸︷︷︸carrier
+PX
2
• The presence of a0 makes envelope detection possible, but
requires extra power ofa202 corresponding to the carrier
• In DSB-SC, we eliminate the a0:
We transmit only the sidebands, and suppress the carrier
1. Multiplying received signal by cos(2πfct) gives
v(t) = x(t) cos2(2πfct) =x(t)
2︸︷︷︸low freq.
+x(t) cos(4πfct)
2︸ ︷︷ ︸high freq.
2. Low-pass filter eliminates the high-frequency component
Ideal low-pass filter has H(f ) = constant for −W ≤ f ≤ W ,and zero otherwise
14 / 32
Properties of DSB-SC
sdsb-sc(t) = x(t) cos(2πfct)
Sdsb-sc(f ) =1
2(X (f + fc) + X (f − fc))
• Bandwidth of DSB-SC is Bdsb-sc = 2W , same as AM
• Power of DSB − SC is Pdsb-sc = PX2
(follows from AM power calculation)
• DSB-SC requires less power than AM as the carrier is nottransmitted
• But DSB-SC receiver is more complex than AM !
We assumed that receiver can generate locally generate afrequency fc sinusoid that is synchronised perfectly in phaseand frequency with transmitter’s carrier
• Effect of phase mismatch at Rx is explored in Examples paper
15 / 32
Single Sideband Suppressed Carrier (SSB-SC)DSB-SC transmits less power than AM. Can we also savebandwidth?
• x(t) is real ⇒ X (−f ) = X ∗(f )
⇒ Need to specify X (f ) only for f > 0• In other words, transmission of both sidebands is not strictly
necessary: we could obtain one sideband from the other!
−fc
Sssb−sc(f)
fc +W−fc −W fc
f0 W−W
X(f)
Upper Sideband
• Bandwidth is Bssb-sc = W , half of that of AM or DSB-SC!
• Power is is Pssb-sc = PX4 , half of DSB-SC
16 / 32
Summary: Amplitude Modulation
−fc fc −W
SAM(f)
f
fc +W−fc +W−fc −W fc
0 W−W
X(f)
−fc fc −W
Sdsb−sc(f)
fc +W−fc +W−fc −W fc
−fc
Sssb−sc(f)
fc +W−fc −W fc
Information signal
AM
DSB-SC
SSB-SC
17 / 32
You can now do Questions 1–5 onExamples Paper 8.
18 / 32
Frequency Modulation (FM)In FM, the information signal x(t) modulates the instantaneousfrequency of the carrier wave.
The instantaneous frequency f (t) is varied linearly with x(t):
f (t) = fc + kf x(t)
This translates to an instantaneous phase θ(t) given by
θ(t) = 2π
∫ t
0f (u)dt = 2πfct + 2πkf
∫ t
0x(u)du
The modulated FM signal
sFM(t) = Ac cos(θ(t)) = Ac cos(2πfct + 2πkf
∫ t
0x(u)du
)
• Ac is the carrier amplitude
• kf is called the frequency-sensitivity factor
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ExampleWhat information signal does this FM wave correspond to?
0 1 2 3 4 5 6 7 8 9 10−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
s FM
(t)
(a) a constant, (b) a ramp, (c) a sinusoid, (d) no clue20 / 32
FM DemodulationAt the receiver, how do we recover x(t) from the FM wave?(ignoring effects of noise)
sFM(t) = Ac cos(2πfct + 2πkf
∫ t
0x(u)du
)
The derivative is
dsFM(t)
dt= −2πAc [fc + kf x(t)] sin
(2πfct + 2πkf
∫ t
0x(u)du
)
• The derivative is a passband signal with amplitude modulationby [fc + kf x(t)]
• If fc large enough, we can recover x(t) by envelope detection
of dsFM(t)dt !
• Hence FM demodulator is a differentiator + envelope detector
• Differentiator: ddt
F−→ j2πf (frequency response). SeeHaykin-Moher book for details on how to build a differentiator