Contents
Phase and angle Modulation Comparison
Effect of Frequency multiplication on the modulated signal
Effect of frequency multiplication of the modulating signal on the modulated signal
Bandwidth using Schwartz Curve
Bessel function of order n and first kind
Transmission of bitstreams using Mary multilevel waveforms
Statistical Properties of Random Variables
Binary Baseband Signal Detection
Autocorrelation & Power Spectral Density
Noise in Communication Systems
Baseband signal transmission receiver containing crosscorrelator.
Probability of error calculation is the same as that of receivers with correlator
BBS through Bandlimited Channels
Communication channel is a Filter
1. Conventional AM
·
· ,which means
· is the index of modulation
· Bandwidth=2W
· W is the bandwidth of
· Power in the modulated signal =
{ This relation is applicable when the modulating signal x(t) is zeromean }
· Total Power :
· Current Relation :
· Modulation Efficiency =
· SNR_{O} =
· Percentage Modulation(%) =
{ This relation is applicable when the modulation is symmetrical }
· Amplitude Modulator for generation.
2. DSBAM
·
· Bandwidth= 2W
· Power in the modulated signal =
· SNR_{O} =
· Balanced Modulator for generation
3. SSBAM
·
· Bandwidth = W
· Power in the signal =
· SNR_{O}
· Methods of generation
Modulating Signal/Message Signal 

Carrier Signal 

Modulated Signal/Transmitted Signal 

Power in the modulating signal 

Power in the carrier signal 

Power in each sideband 

Power in the transmitted signal 

Modulation Index 

Modulation Efficiency 

Modulation Percentage (%) by applying sawtooth wave to the horizontal plate and modulated wave to the vertical plates of an oscilloscope


Modulation Percentage (%) by applying modulated signal to vertical plate and modulating signal to horizontal plate of an oscilloscope


Time constant of Envelope Detector:
If and are the time period of modulating and carrier wave respectively then for faithful reproduction of message signal. the time constant of RC circuit should be less than and greater than . 

1. Need for frequency translation
a. To achieve practical lengths of antennas.
b. Reduces the ratio of highest to the lowest frequency band in the signal.
2. Method of frequency translation
a. Multiplication with sinusoidal time signal of higher frequency than the base signal.
3. Recovery of the baseband signal
a. Multiplication with sinusoidal time signal followed by lowpass filtering. [this method is applicable only if the phase of the signal is constant].
4. Multiplier and Mixer
Mixer is about addition
Multiplier is about product
5. Balanced Modulator and Amplitude Modulator.
Amplitude Modulator is a multiplier and is a nonlinear device (eg. ClassC amplifiers)
Balanced modulator is a pair of multipliers followed by an adder. Its output is multiplication of input only (none of the input appear individually at the output).
Hilbert Transform(HT)
In the field of signal processing:
1. Hilbert Transform of a signal introduces phase shift of 90^{0} in all the signal components.
2. Hilbert Transform gives analytic representation of a signal
HT of a signal is defined as
Instantaneous frequency
1. Phase Modulated signal
·
· Maximum frequency deviation =
It is the positive maximum deviation of the phase of the modulated signal from the phase of the carrier.
· Modulation index for sinusoidal message signal =
· Modulation index for nonsinusoidal message signal
2. Frequency Modulated signal
·
· Instantaneous phase is
· Instantaneous frequency is
· Maximum frequency deviation when the message signal is
It is the positive maximum deviation of the frequency of the modulated signal from the frequency of the carrier.
· Carrier Swing
It is the maximum deviation of the frequency of the modulated signal from the frequency of the carrier.
· Modulation index for sinusoidal message signal =
· Modulation percent
· Bandwidth =
· Bandwidth can also be calculated using Schwartz Curve
· Modulation index for nonsinusoidal message signal
· If message signal is sinusoidal then modulated signal can be expressed as
where
3. Frequency Spectrum of Angle Modulated Signal
4. Methods of FM Generation
1. Parameter Variation Method
2. Armstrong Method (uses frequency multiplication)
5. Frequency Multiplication and its effect on modulation index and frequency deviation
· ntimes frequency multiplication increases the modulation index by n times
· ntimes frequency multiplication increases the frequency deviation by n times
Parameters 
Phase Modulation

Frequency Modulation

Modulated Signal 


Instantaneous Phase 


Instantaneous Frequency



Deviation 
Phase deviation 
Frequency Deviation 
Modulation Index 


Bandwidth 


Narrowband 


Parameters 
PM 
FM 
Deviation 
Phase deviation 
Frequency deviation 
Modulation index 
Modulation index 
Modulation index 
Bandwidth 
Bandwidth 
Bandwidth 
Parameters 
PM 
FM 
Deviation 
Phase deviation unchanged 
Frequency deviation 
Modulation index 
Modulation index unchanged 
Modulation index 
Bandwidth 
Bandwidth unchanged 
Bandwidth unchanged 
Say modulation index = 4 then from the curve
Criteria 
Type of FM 
Bandwidth 
modulation index 
Wideband FM 
Bandwidth 
modulation index 
Narrowband FM 
Bandwidth 
Narrow band is comparable to Amplitude modulation
If the modulating signal is not sinusoidal then it falls under the category of arbitrary modulation of carrier . For such system modulation index is called Deviation ratio
is the bandwidth of modulating signal.
Bandwidth of the modulated signal
SNR FM 

(W/Hz) 
SNR PM 

(W/Hz) 
SNR AM 

(W/Hz) 
·
· if
· if
Let us start with a signal whose Fourier transform is and is assumed band limited .
OR

The Fourier transform of is given below. The first expression could be used to prove that is necessary for faithful reproduction of from
OR

Regeneration of from the Fourier Transform of the
can be obtained from if was sampled at Nyquist rate Thus

Substituting from the second expression in gives
Taking Inverse Fourier transform gives

In quantization process actual value of the sampled signal is replaced by a quantized value.
Type of Modulation 
Quantization Error (mean squared) 
PCM 

Delta 

1. Let analog signal vary form to
2. Divide the signal swing into intervals (here )
3. Hence Step size of each interval is
4. In any interval the value of the signal is approximated to midpoint of that interval.
5. SNR =
If quantization error is to be less than voltage, where is some fraction, then the minimum bits required is 

Line Coding 

· If signals with equal bandwidth is transmitted using TDM then the sampling frequency is
· Let us say that a stream of 0's and 1's is spewed into the channel. If the time period of the pulse used to transmit and is then the bitrate is
So the channel gets pulse at time and so on.
The receiver also checks for pulses at and so on. If ISI (Inter symbol interference) occurs then receiver gives error.
ISI occurs if pulse spread (which they do in physical channel) and at the time of sampling at receiver they give false output as shown in the figure.
If the pulse shape is perfect sinc with zero crossing at then at receiver ISI is zero, but such pulses have ideal frequency response hence not realizable.
Rule of Thumb: A transmission with bit rate is successfully detected without ISI if the minimum channel bandwidth is . The signal used in such a transmission is called Nyquist pulse.
Using Raised cosine pulse, rolloff , the minimum channel bandwidth required will be

Unipolar NRZ 


Unipolar ZR 


Bipolar NRZ 


Bipolar ZR 


AMI 

Analog signal with amplitude range and maximum frequency is PCM encoded using bits per sample. The signal is sampled at a rate of and a multilevel signal is used for transmission.
Number of Levels (L) 

Step Size (S) 

Average Quantization Noise Power 

SNR (db) 

SNR (in terms of ) (db) 

SNR (in terms of system bandwidth) (db) 

System bandwidth 
is the number of bits in multilevel signalling that is used to transmit PCM encoded analog signal 
Nyquist rate of sampling a PCM Encoded signal 

Bit rate of PCM Encoded signal 

Channel Bandwidth 

Analog signal with bandwidth is sampled at and delta modulated with step size and a post construction filter with bandwidth is used
Maximum quantization error 

Step size to avoid Slope overload 

SNR (db) with postconstruction filter 

SNR (db) taking without postconstruction filter 

Random Variables are functions that map events (sample points) to real valued numbers.
A random variable is a which maps sample points to a real valued number. Simply put
· In the context of random variable sample space is also known as 'domain'.
· The values taken by is called the 'range of '.
To calculate Probability of a random variable is it necessary to have either its cdf or pms or pdf.


Properties of 
1. 2. 3. 4. 5. 6. 
Properties of 
1.
2.
3. 
Properties of 
1. 2. 3. 
Conditional Probability 
If or then event and are statistically independent 
Total Probability : Let there be a sample space of mutually exclusive and exhaustive events i.e. .Let there be an event B in S , then is 

Bayes Theorem


Let and represent sent and receive respectively and it is given

Probability that a was sent 

Probability that a 0 was received 

Probability that a was sent 

Probability that a was received 








Calculate 
Solution 
P( 
) ))) 

)) 












Probability of error 
= 
Probability of zero error 
= 
Binomial Distribution
Discrete RV function 

If and then binomial distribution can be approximated to Poisson Distribution with the following substitution

Poisson Distribution
Discrete RV function



Normal Distribution
Continuous RV function


is complementary error function

Exponential Distribution
Continuous RV function 


Property 
Using probability Mass Function 
Using probability density function 
Mean 


nMoment 


Variance 


Correlation 


Covariance 


If then , and are said to be uncorrelated.
If then and are independent.
If and are independent then but the converse is not necessarily true.
If and are independent then
Moment Generating function : . The moment is define as
Probability of Error 
1.
2.
is the error complimentary function


Gaussian probability density when was sent. Gaussian probability density when was sent. Threshold value for detection
The detector outputs if The detector outputs if
Since is a R.V its exact value is undetermined. However probability that or can be determined using likelihood densities. 
What is the probability that was transmitted but was detected 

What is the probability that was transmitted but was detected 

What is the probability of bit error 

Evaluation of threshold value
are mean values of and is the variance of AWGN 


1. is a random process 2. is a random variable 3. is a sample function 4. All sample functions is called an ensemble
The values that a random process can take is called statespace. It could be discretestate or continuousstate.
The time line of a random process is called index set. It could be discreteparameter or continuous parameter corresponding to discrete or continuous time respectively.

Density function of Random process : The order density function of RP is 

Ensemble Average (mean) of 

Autocorrelation 

Covariance 

Strict Sense Stationary 
A random process is SSS if its density is dependent on time difference only and not on the origin of time

Wide Sense Stationary 
A random process is WSS if its mean is constant and autocorrelation is dependent on time difference only
1. 2. 
Timeaverage mean 

Timeaverage Autocorrelation 

Ergodic Random Process 
A random process is called ergodic if ensemble average equals timeaverage 
Autocorrelation 
Crosscorrelation 
Autocovariance 
Crosscovariance 
1. 2. 3. 4.

1. If implies that and are orthogonal 2.

1.

1. 
Passing a random process through an LTI system
Properties of Autocorrelation Function 
Properties of Power Spectral Density (watts/Hz) 
1. 2. 3. 4. 
1. 2. 3. 4. 
Valid Autocorrelation Function 
Invalid Autocorrelation Function 




Valid Power Spectral Density

Invalid Power Spectral Density










Modulation System 
General Arrangement of Receiver System 
Base band System 

Amplitude Modulated System 

Angle Modulated System 

Noise in Baseband Communication System
1. Receiver is an Ideal Lowpass filter with bandwidth 2. Message signal is also bandlimited to 3. For sake of simplicity substitute

Synchronous Detection


Noise in Amplitude Modulated DSB System
1. Receiver is a combination of Ideal Bandpass filter with bandwidth , demodulator and an Ideal Lowpass filter with bandwidth 2 2. Message signal is also bandlimited to 2 3. For sake of simplicity substitute
Bandpass filter is also known as predetection filter.

Synchronous Detection


Noise in Amplitude Modulated SSB System
1. Receiver is a combination of Ideal Bandpass filter , demodulator and an Ideal Lowpass filter with bandwidth 2. Message signal is also bandlimited to 3. For sake of simplicity substitute

Synchronous Detection


Noise in Normal Amplitude Modulated System 
Synchronous Detection
Envelope Detection

The phenomenon where degrades very rapidly using envelope detection is called threshold effect.
For sinusoidal signal and 100% modulation

SNR in PM 


SNR in PM 


Assuming that is a random process message signal with average power . The SNR of some of the common modulation schemes
Normal AM Signal with message bandwidth (Hz) 

After synchronous demodulation

DSB AM Signal with message bandwidth 

After Synchronous demodulation

SSB(Lower Side band) AM Signal with message bandwidth
OR
SSB(Upper Side band) AM Signal with message bandwidth 
OR

After Synchronous demodulation

PM Signal 


FM Signal 

* The complicated expression for is because it is given by the derivate of the quadrature component of the noise , . 
The transfer function of a matched filter


A matched filter produces maximum SNR at the output for a given signal at its input 

· Lets see what happens step by step:
a. s_{i}(t) is transmitted [ it could be s_{0}(t) or s_{1}(t)].
b. s_{i}(t) enters the channel and get corrupted by noise. The source of noise could be anything. Mathematically this noise is modeled as thermal noise. All that we know about this noise is its powerspectral density. Let say the corrupted signal is r(t).
c. r(t) hits the correlator.
d. Correlator integrates and its output is sampled at T_{b}.
Case1: s_{0}(t) is transmitted
· r(t) is a random process because n(t) is a Gaussian random process.
· r(t) after passing through correlator and sampler gives a random variable.
· r_{0} is a random variable and r_{1} is also a random variable.
mean(
This is the point
where things get a little complicated. I cannot express
like this [ So what I am looking at is .
Autocorrelation is defined just for this kind of expression.
Finally
· Mean (r_{0})=Energy of s_{0}(t) ;Var(r_{0})=Energy of s_{0}(t)
· Mean (r_{1})=0 ;Var(r_{1})=Energy of s_{1}(t)
· Probability density of r_{0} when s_{0}(t) is transmitted is
· Probability density of r_{1} when s_{0}(t) is transmitted is
· Calculation of probability of error: If is transmitted and the output of the detector is then an error has occurred. Detector would make this mistake only when the value of is greater than [+ which implies that
·
· is again a random variable. Now we have to come up with ways to finds its probability density in order to evaluate
· Mean=mean()mean()=0
·
(Since both the signals here have equal energy)
·
Greater the magnitude of the argument of error function Q the smaller its value. So by increasing the energy of the signal probability of error could be reduced.
Case2: s_{1}(t) is transmitted
· Probability density of r_{0} when s_{0}(t) is transmitted is
· Probability density of r_{1} when s_{0}(t) is transmitted is
· Baseband signal transmission receiver containing matchedfilter.
Case1: s_{0}(t) is transmitted
· 4amplitude signal
Signal 

Waveforms 
Probability of error in terms of Error function Q 
Signal Energy per bit 

Binary Orthogonal 




No of correlators in the receiver =2 
Binary Antipodal 

or

3db better than orthogonal signal for the same signal energy 

No of correlators in the receiver =1 
4amplitude signals (onedimensional signals) 
g(t) is unit energy pulse



Each symbol has different engery.
Average symbol energy =
Average bit energy 
No of correlators in the receiver =1 
mamplitude signals (onedimensional signals) 
g(t) is unit energy pulse



Each symbol has different engery.
Average symbol energy =
Average bit energy 
No of correlators in the receiver = 1 
Orthogonal Multidimensional signal 


= average probability of bit error.
= average probability of symbol error. 
No of correlators in the receiver = m 

Biorthogonal 
No of signals= m
are orthogonal
) are antiopdal 


No of correlators in the receiver 
· Expression for a PAM signal s(t) with symbol interval T
· The point to understand here is that is a random variable and is a sample function of a random process S(t). To determine the spectral characteristics of random process we must evaluate its power spectrum and here autocorrelation comes into the picture.
Symbol source 
Transmitter Pulse

Spectrum of symbol source 
Fourier Transform of Transmitter pulse 
Spectrum

Plot of spectrum 






is uncorrelated with autocorrelation





T=1

is uncorrelated with autocorrelation





T=1

Frequency response of some communication channels like telephone channels and some radio channels can be modeled as
· Group delay is defined as
· Group delay is also known as envelope delay.
· Group delay not constant means delay distortion.
· Delay distortion causes intersymbol interference(ISI).
· Use of equalizers or filters reduce ISI.
Some channels like shortwave propagation through ionosphere, mobile cellular radio and tropospheric scatter are modeled using scattering function.