What is Angle Modulation?
In angle modulation, information is embedded in the angle of the carrier. We define the angle of a modulated carrier by the argument of... st Ac cos cos t
Phasor Form
In the complex plane we have t=3 Phasor rotates with nonuniform speed t=1
t=0
Angular Velocity
Since phase changes nonuniformly vs. time, we can define a rate of change
i
d i (t ) dt
This is what we know as frequency d i 2 f c st Ac cos2 f ct c t dt i
Instantaneous Frequency
We are used to signals with constant carrier frequency. There are cases where carrier frequency itself changes with time. We can therefor talk about instantaneous frequency defined as f i t
1 d i t 2 dt
Examples of Inst. Freq.
Consider an AM signal d i st 1 km( km(t )cos2 f c t c 2 f c t dt Here, the instantaneous frequency is the frequency itself, which is constant i
Impressing a message on the angle of carrier
There are two ways to form a an angle modulated signal. – Embed it in the phase of the carrier Phase Modulation(PM) – Embed it in the frequency of the carrier Frequency Modulation Modulation(FM) (FM)
Phase Modulation(PM)
In PM, carrier angle changes linearly with the message
st Ac cos cos i t Ac cos 2 f ct k pmt
Where – 2πf c=angle of unmodulated carrier – kp=phase sensitivity in radians/volt
Frequency Modulation
In FM, it is the instantaneous frequency that varies linearly with message amplitude, i.e. fi(t)=fc+kfm(t)
FM Signal
We saw that I.F. is the derivative of the phase 1 d i t f i t 2 dt
Therefore, t
i t 2 f c t 2 k k f mt 0
t st Ac cos cos2 f c t 2 k k f m(t ) dt 0
FM for Tone Signals
Consider a sinusoidal message m(t ) Am cos2 f mt The instantaneous frequency corresponding to its FM version is f i t f c
k f m(t )
f c resting frequency frequency
k f Am cos2 f mt
Illustrating FM 1
Inst.frequency Moves with the Message amplitude
FM message
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1 0
0. 01
0. 02
0.03
0.04
0.05
0.06
0. 07
0. 08
0.09
0. 1
Frequency Deviation
Inst. frequency has upper and lower bounds given by f i t f c f cos2 f mt where f f frequency deviation k f Am then f i max f c f f i min f c f min
FM Modulation index
The equivalent of AM modulation index is which is also called deviation ratio . It quantifies how much carrier frequency swings relative to message bandwidth
f
f
ba seband seband
tone
or W f m
Example:carrier swing
A 100 MHz FM carrier is modulated by an audio tone causing 20 KHz frequency deviation. Determine the carrier siwng and highest and lowest carrier frequencies KHz f 20 KHz fre fr eq uency swing uency swing 2 f 40 KHz KHz fre fr eq uency range: range : f high f low
MHz Hz 20 KHz KHz 100.02 M MHz Hz 100 M Hz 20 KHz KHz 99.98 M MHz Hz 100 M
Example: deviation ratio
What is the modulation index (or deviation ratio) of an FM signal with carrier swing of 150 KHz when the modulating signal is 15 KHz? f
150
75 KHz KHz
2 f 75
f m
15
5
Myth of FM
Deriving FM bandwidth is a lot more involved than AM FM was initially thought to be a bandwidth efficient communication because it was thought that FM bandwidth is simply 2 f By keeping frequency deviation low, we can use arbitrary small bandwidth
FM bandwidth
Deriving FM bandwidth is a lot more involved than AM and it can barely be derived for sinusoidal message There is a graphical way to illustrate FM bandwidth
Piece-wise approximation of baseband
Look at the following representation
Baseband bandwidth =W
1/2W
Corresponding FM signal
FM version of the above is an RF pulse for each square pulse. The frequency of the kth RF pulse at t=tk is given by the height of the pulse. i.e. f i f c
k f mt k
Range of frequencies?
We have a bunch of RF pulses each at a different frequency. Inst.freq corresponding to square pulses lie in the following range f i max
f c
k f mmax
f i min
f c
k f mmin mmin
mmax
A look at the spectrum
We will have a series of RF pulses each at a different frequency. The collective spectrum is a bunch of sincs lowest
highest
f 4W
So what is the bandwidth?
Measure the width from the first upper zero crossing of the highest term to the first lower zero crossing of the lowest term highest lowest
f
Closer look
The highest sinc is located at fc+kfmp Each sinc is 1/2W wide. Therefore, their zero crossing point is always 2W above the center of the sinc.
2W
f
Range of frequenices lowest
highest
f
Above range lies
FM bandwidth
The range just defined is one expression for FM bandwidth. There are many more! BFM=4W+2kfmp Using =∆f/W with ∆f=kfmp BFM=2( +2)W
Carson’s Rule
A popular expression for FM bandwidth is Carson’s rule. It is a bit smaller than what we just saw BFM=2( +1)W
Commercial FM
Commercial FM broadcasting uses the following parameters – Baseband;15KHz – Deviation ratio:5 – Peak freq. Deviation=75KHz BFM=2(+1)W=2x6x15=180KHz
Wideband vs. narrowband FM
NBFM is defined by the condition – ∆f<
BFM=2W
– This is just like AM. No advantage here
WBFM is defined by the condition – ∆f>>W
BFM=2 ∆f
– This is what we have for a true FM signal
Boundary between narrowband and wideband FM
This distinction is controlled by – If >1 --> WBFM – If <1-->NBFM
Needless to say there is no point for going with NBFM because the signal looks and sounds more like AM
Commercial FM spectrum
The FM landscape looks like this carrier
FM station A
FM station B
150 KHz 200 KHz
25KHz guardband
FM station C
FM stereo:multiplexing
First, two channels are created; (left+right) and (left-right) Left+right is useable by monaural receivers Left channel + +
Right channel
+
-
mono
Subcarrier modulation
The mono signal is left alone but the difference channel is amplitude modulated with a 38 KHz carrier Left channel + Right channel
Composite baseband
mono
+
+ +
DSB-SC f sc=38 kHz -
fsc= 38KHz
freq divider
Stereo signal
Composite baseband signal is then frequency modulated Composite baseband
Left channel
+
mono
+
+
Right channel +
DSB-SC f sc=38 kHz
fsc= 38KHz
freq divider
FM transmitter
Stereo spectrum
Baseband spectrum holds all the information. It consists of composite baseband, pilot tone and DSB-SC spectrum Left+right
DSB-SC
19 KHz 15 KHz
38 KHz
Stereo receiver
First, FM is stripped, i.e. demodulated Second, composite baseband is lowpass filtered to recover the left+right and in parallel amplitude demodulated to recover the left-right signal Left+right
DSB-SC
19 KHz 15 KHz
38 KHz
Receiver diagram + lowpass filter(15K)
Left+right
+
left
+
coherent detector 15 KHz 19 KHz 38 KHz
bandpass at 38KHz
X
lowpass
right
- +
+
FM receiver PLL X
lowepass
Divide 2
VCO
Subsidiary communication authorization(SCA)
It is possible to transmit “special programming” ,e.g. commercial-free music for banks, department stores etc. embedded in the regular FM programming
Such programming is frequency multiplexed on the FM signal with a 67 KHz carrier and 7.5 KHz deviation
SCA spectrum
Left+right
DSB-SC SCA signal
19 KHz 15 KHz
38 KHz
59.5
67
74.5
f(KHz)
FM receiver
FM receiver is similar to the superhet layout
RF
mixer
LO
IF
limiter
AF power amp
Discriminator
deemphasis
Frequency demodulation
Remember that message in an FM signal is in the instantaneous frequency or equivalently derivative of carrier angle t st Ac cos cos2 f ct 2 k k f m(t ) dt 0
st Ac
t 2 f c 2 k k f mt sin2 f c t 2 k k f m(t )dt
Do envelope detection on s’(t)
Receiver components:RF amplifier
AM may skip RF amp but FM requires it FM receivers are called upon to work with weak signals (~1 V or less as compared to 30 V for AM) An RF section is needed to bring up the signal to at least 10 to 20 V before mixing
Limiter
A limiter is a circuit whose output is constant for all input amplitudes above a threshold Limiter’s function in an FM receiver is to remove unwanted amplitude variations of the FM signal Limiter
Limiting and sensitivity
A limiter needs about 1V of signal, called quieting or threshold voltage, to begin limiting When enough signal arrives at the receiver to start limiting action, the set quiets, i.e. background noise disappears Sensitivity is the min. RF signal to produce a specified level of quieting, normally
Sensitivity example
An FM receiver provides a voltage gain of 200,000(106dB) prior to its limiter. The limiter’s quieting voltage is 200 mV. What is the receiver’s sensitivity? What we are really asking is the required signal at RF’s input to produce 200 mV at the output 200 mV/200,000= 1 V->sensitivity
Discriminator
The heart of FM is this relationship f i(t)=f c+k f m(t)
What we need is a device that linearly f is at the IF frequency follows inst. frequency Of 10.7 MHz carrier
Disc.output
-75 KHz
+75 KHz f carrier
Deviation limits
f
Examples of discriminators
Slope detector - simple LC tank circuit operated at its most linear response curve This setup turns an FM signal into an AM
output
f c
f o
f
Phase-Locked Loop
PLL’s are increasingly used as FM demodulators and appear at IF output fin
Phase comparator
Error signal
Lowpass filter
Output proportional to Difference between f in and f vco
Control signal:constant When f in=f vco
f vco
VCO
VCO input
PLL states
Free-running – If the input and VCO frequency are too far apart, PLL free-runs
Capture – Once VCO closes in on the input frequency, PLL is said to be in the tracking or capture mode
Locked or tracking – Can stay locked over a wider range than was necessary for capture
PLL example
VCO free-runs at 10 MHZ. VCO does not change frequency until the input is within 50 KHZ. In the tracking mode, VCO follows the input to ±200 KHz of 10 MHz before losing lock. What is the lock and capture range? – Capture range= 2x50KHz=100 KHz – Lock range=2x200 KHz=400 KHz
Advantages of PLL
If there is a carrier center frequency or LO frequency drift, conventional detectors will be untuned PLL, on the other hand, can correct itself. PLL’s need no tuned circuits
output
If f c drifts detector has no way of correcting itself
Slope detector f c
f o
f
Zero crossing detector FM
Hard
limiter
Zero Crossing detector
Multivibrator
Averaging circuot
Output
FM input
Hard limiter
ZC detector
multiV
Averaging circuit
more frequent ZC’s means higher inst freq in turn means Larger message amplitudes
NOISE IN ANALOG MODULATION
AMPLITUDE MODULATION
Receiver Model
The objective here is to establish a relationship between input and and output SNR of an AM receiver Modulated signal s(t)l BPF
detector
filter
output
BT=2W
Noise n(t)
-f c
f c
Establishing a reference SNR
Define “channel” SNR measured at receiver input
(SNR)c=avg. power of modulated signal/ avg. noise power in the message bandwidth
Noise in DSB-SC Receiver
Tuner plus coherent detection DSB-SC
BPF
x(t)
LPF
v(t)
s(t)
n(t)
Cos(2πfct)
st Ac m(t ) co cos 2 f c t 2 2 s2 t avg . power power Ac 2 m ( t ) / 2 Ac P / P / 2 P avg avg. me message ssage power
Receiver input SNR
Also defined as channel SNR: ( SNR SNR)c
Ac2 P / P / 2
WN o nois nois e power in the t he message bandwidth
Flat noise spectrum:white noise
No/2
Noise power=hatched area
-W
W
2
Ac P 2WN o
Output SNR
Carrying signal and noise through the rest of the receiver, it can be shown that output SNR comes out to be equal to the input. Hence SNRo 1 SNRc
Therefore, any reduction in input SNR is linearly reflected in the output
(SNR)o for DSB-AM
Following a similar approach, 2 SNRo k P 1 2 SNRc 1 k P
k : k : AM modulation odulation index P : P : avg avg. message message power
Best case is achieved for 100% modulation index which, for tone modulation, is only 1/3
DSB-AM and DSB-SC noise performance
An AM system using envelope detection needs 3 times as much power to achieve the same output SNR as a suppressed carrier AM with coherent detection This is a result similar to power efficiency of the two schemes
Threshold effect-AM
In DSB-AM (not DSB-SC) there is a phenomenon called threshold effect This means that there is a massive drop in output SNR if input SNR drops below a threshold For DSB-AM with envelope detection, this threshold is about 6.6 dB
NOISE IN ANALOG MODULATION
FREQUENCY MODULATION
Receiver model FM s(t)
BFP
Limiter
FM detector
LPF (W)
n(t)
Noisy FM signal at BPF’s output is x t st n(t )
Ac cos 2 f ct t r (t ) co cos 2 f c t t noise
where
t m(t )dt
Phasor model
We can see the effect of noise graphically
(t)
(t)
(t)
reference The angle FM detector will extract
Small noise
For small noise, it can be approximated that the noise inflicted phase error is =[r⁄Ac]Sin( So the angle available to the FM detector is + FM Detector computes the derivative of this angle. It will then follow that...
FM SNR for tone modulation
Skipping further detail, we can show that for tone modulation, we have the following ratio SNR SNRo 3 2 SNR SNRc 2
SNR rises as power of 2 of bandwidth; e.g. doubling deviation ratio quadruples the SNR Bandwidth-SNR exchange
Comparison with AM
In DSB-SC the ratio was 1 regardless. For commercial FM, =5. Therefore, (SNR)o/(SNR)c=(1.5)x25=37.5 Compare this with just 1 for AM
Capture effect in FM
An FM receiver locks on to the stronger of two received signals of the same frequency and suppresses the weaker one Capture ratio is the necessary difference(in dB) between the two signals for capture effect to go into action Typical number for capture ratio is 1 dB
Normalized transmission bandwidth With all these bandwidths numbers, it is good to have a normalized quantity. Define normalized bandwidth=Bn=BT/W Where W is the baseband bandwidth
Examples of Bn
For AM: Bn=BT/W=2W/W=2
For FM Bn=BT/W~2 to 3 For =5 in commercial FM, this is a very large expenditure in bandwidth which is rewarded in increased SNR
Noise/bandwidth summary
AM-envelope detection
SNRo Bn
2
2 2
2
SNRc
Noise/bandwidth summary
DSB-SC/coherent detection (SNR)o=(SNR)c Bn=2
SSB (SNR)o=(SNR)c Bn=1
Noise/bandwidth summary
FM-tone modulation and =5 (SNR)o=1.5 2(SNR)c=37.5 (SNR)c Bn~16 for =5
Preemphasis and deemphasis
High pitched sounds are generally of lower amplitude than bass. In FM lower amplitudes means lower frequency deviation hence lower SNR. Preemphasis is a technique where high frequency components are amplified before modulation Deemphasis network returns the baseband to its original form
Pre/deemphasis response
Flat up to ~500Hz, rises from 500-15000 Hz
17dB preemphasis +3dB
-3dB deemphasis -17dB 500 Hz
2120 Hz
15KHz
Deemphasis circuit Is between the detector And the audio amplifier
Suggested homework
3.41 5.3 5.7
