Lecture 21: Frequency and phase modulationDANIEL WELLER

TUESDAY, APRIL 9, 2019

AgendaLimitations of amplitude modulation

Sinusoid parameters: frequency and phase

Frequency modulation and frequency shift keying

Phase modulation and phase shift keying

Quadrature phase shift keying (QPSK)

Broadcast at higher frequencies, FM radio provides higher quality audio than AM radio and is favored for broadcasting music. Above is the first FM radio transmitter, then located in a lab in the Empire State Building in New York City.

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Image credit: Popular Science / Wikipedia

Recall: amplitude modulationRecall amplitude modulation:

An amplitude-modulated signal modifies the carrier amplitude by the scaled message xm(t), plus 1.

What is the frequency relationship?

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𝑥𝐴𝑀 𝑡 = 𝑥𝐶 𝑡 1 + 𝑚𝑥𝑚 𝑡

xAM(t) = AC cos(2fCt) [1 + m cos(2fmt)]

fC fC + fmfC - fm

tone modulation

Limitations of AMCarrier power – significant, but does not convey information

◦ Can use AM-SC with suppressed carrier, but envelope detection no longer possible

Maximum scaling – message signal must be scaled to fit carrier to permit envelope detection◦ Recall: when mxm(t) < -1, the envelope crosses itself

Redundant frequency content◦ Single-sideband AM possible, but requires more

complicated demodulation

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xAM(t) = xC(t)[1+mxm(t)]

m = 1.5

Envelope [1+mxm(t)] when m > 1

Limitations of AMConsider the effect of noise on an AM signal:

◦ What happens where 1+mxm(t) is large?

◦ What about when 1+mxm(t) is close to zero?

What about an interfering signal?

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Limitations of AMWhat happens with noise or interference in the frequency domain?

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noise level (blue)

Revisiting the carrier signalLet us revisit the definition of a sinusoidal carrier signal xC(t):

So far, we considered amplitude modulation, where A becomes a function of the message xm(t):

However, we are not limited to modifying the amplitude A. We can also modify the frequency of the carrier and its phase:

◦ Frequency becomes a function of the message signal versus time

◦ Phase becomes a function of the message signal

◦ Consider: frequency is the time derivative of the argument of the sinusoid

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𝑥𝐶 𝑡 = 𝐴 cos 2𝜋𝑓𝐶𝑡 + 𝜙

𝐴 = 1 +𝑚𝑥𝑚 𝑡

Frequency modulationThe argument θ in the sinusoidal carrier is a function of time:

Then the derivative of θ with respect to t is the carrier frequency:

Instead, let this be a function of the message signal as well:

◦ Let fΔ be a scaling factor for FM, like m for AM (this will turn out to be a function of available bandwidth)

◦ Then, the argument becomes an integral (time t=0 is the beginning of the transmission):

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𝜃 𝑡 = 2𝜋𝑓𝐶𝑡 + 𝜙

𝑑𝜃 𝑡

𝑑𝑡= 2𝜋𝑓𝐶

𝑑𝜃 𝑡

𝑑𝑡= 2𝜋 𝑓𝐶 + 𝑓Δ𝑥𝑚 𝑡

𝜃 𝑡 = 2𝜋 𝑓𝐶𝑡 + 𝑓Δන0

𝑡

𝑥𝑚 𝜏 𝑑𝜏 + 𝜙

Frequency modulationThe amplitude of the carrier remains constant

◦ Consistent noise scaling

The frequency affects how rapidly the sinusoid changes sign: higher frequency, more sign changes per unit time.

◦ Demodulation: track sign change rate

◦ More complicated than amplitude demodulation (envelope detection), but less noise sensitivity since signal is “always on”

Analog frequency modulation (FM) digital frequency shift keying (FSK)◦ Two fixed frequencies: f0 for “0”, f1 for “1”

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Frequency shift keyingConsider a simple binary message xm(t):

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Frequency Shift Keying

Phase modulationFrequency is not the only choice in the argument of the sinusoid for modulation. Also consider modulating the phase φ:

Replace phase with a message signal xm(t):

◦ The message signal can be scaled by a constant factor as well.

What if we take the derivative? Phase modulation becomes a special case of FM:

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𝜃 𝑡 = 2𝜋𝑓𝐶𝑡 + 𝜙

𝜃 𝑡 = 2𝜋𝑓𝐶𝑡 + 𝑥𝑚 𝑡 + 𝜙

𝑑𝜃 𝑡

𝑑𝑡= 2𝜋 𝑓𝐶 +

𝑑𝑥𝑚 𝑡

𝑑𝑡

Phase shift keyingAnalog phase modulation digital phase shift keying (PSK)

◦ Replace analog xm(t) with binary “0” and “1”, so phase jumps between φ0 (usually 0) and φ1 (usually 180 degrees)

◦ This is not differentiable, so it’s not really the same as FSK…

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Phase Shift Keying

Phase Shift Keying

Note jumps at arrows: lines up with xm(t)

Phase shift keyingLet’s take a closer look:

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Phase Shift Keying

Modulation summaryThree methods for modulating binary/digital signals:

◦ Amplitude shift keying: track amplitude zero/one

◦ Frequency shift keying: track zero-crossing rate

◦ Phase shift keying: track phase jumps/discontinuities

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Modulation Summary

Carrier

Information

ASK

FSK

PSK

Advantages and disadvantagesAmplitude modulation (AM/ASK):

◦ Simple design, demodulation (e.g., envelope detection)

◦ Susceptible to noise (especially when transmitting small/zero values)

Frequency modulation (FM/FSK):◦ Amplitude constant, so signal always present and better noise immunity

◦ Need to track zero-crossing rate, not just instantaneous amplitude

◦ Frequencies spread out, requiring more bandwidth

Phase modulation (PSK):◦ Similar advantages to FM/FSK, but no frequency spreading!

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Modern cellular modemsCellular modems transmit/receive voice/data from phone over cellular network

◦ Responsible for encoding and modulating signal according to network type

Early modems: mainly amplitude modulation (ASK)

Modern modems: combination of PSK, ASK (can encode information in amplitude and phase changes simultaneously; harder to encode in both FSK and PSK)

◦ Modern modems actually go beyond PSK to multibit encoding

◦ Let’s investigate phase modulation a bit more carefully…

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5G modem will require its own chip for now; current 4G LTE use integrated “system-on-chip” solutions

Phase shift keying: decoding180 degree phase shift (a “1”) introduces a jump discontinuity in the carrier:

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180o phase shift

Phase shift keying: decodingBut a “0” doesn’t create any shift.

Transmitter, receiver need to agree on a bit rate.

Instead of encoding “0” and “1” directly, we use differential encoding:

◦ A change is a “1”, no change is a “0”

◦ Requires we know how it starts

◦ If we miss a bit, we’re in trouble…

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0o phase shift

Phase shift keying - decodingWith differential coding, another “1” indicates a change from 1 0:

Can we do better? ◦ 180 degree jumps are big, easy to detect

◦ 90 degree jumps go between ±1 and 0, also relatively easy to detect

◦ Let’s allow 90, 180, 270 degree jumps: quadrature phase shift keying (QPSK)

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180o phase shift

Quadrature phase shift keyingDefine a “dibit” (two bit message) to encode as 0, 90, 180, 270 degree phase shifts:

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180o

QPSK Dibit encoding

0o

270o

90o

00

0110

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These dibit assignments to each phase jump are arbitrary…

Quadrature phase shift keyingA little more complicated, but twice the information transfer!

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0o

01

90o

00

180o

10 270o

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Example of QPSKWhat is the bit rate for this example?

= (2 bits) / (0.01 seconds) = 200 bits/sec

Regular encoding:

11011000

Differential encoding (1 = change, 0 = no change):

10010000

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270o

11

0o

01

180o

10

90o

00

Your turnSketch carrier modulated using QPSK for binary signal “10010011” using regular (not differential) encoding, assuming each dibit is modulated over a full period of xC(t)

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180o

QPSK Dibit encoding

0o

270o

90o

00

0110

11

xQPSK(t)

tT 2T 3T 4T 5T

AnnouncementsNext class: Number systems and binary arithmetic

ECE 2066: Lab 7 today; also Lab 6 due via Collab

Homework 7 due this Thursday

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