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This document covers topics related to analogue communication techniques, including spectral analysis of signals, amplitude modulation techniques, angle modulation, mathematical representation of noise, noise in AM and FM systems. The document also includes mathematical formulas and properties of Fourier transform, response of a linear system, and transfer function. intended for students studying communication systems and related fields.
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Module-I (12 Hours)
Spectral Analysis: Fourier Series: The Sampling Function, The Response of a linear System, Normalized Power in a Fourier expansion, Impulse Response, Power Spectral Density, Effect of Transfer Function on Power Spectral Density, The Fourier Transform, Physical Appreciation of the Fourier Transform, Transform of some useful functions, Scaling, Time-shifting and Frequency shifting properties, Convolution, Parseval's Theorem, Correlation between waveforms, Auto-and cross correlation, Expansion in Orthogonal Functions, Correspondence between signals and Vectors, Distinguishability of Signals.
Module-II (14 Hours)
Amplitude Modulation Systems: A Method of frequency translation, Recovery of base band Signal, Amplitude Modulation, Spectrum of AM Signal, The Balanced Modulator, The Square law Demodulator, DSB-SC, SSB-SC and VSB, Their Methods of Generation and Demodulation, Carrier Acquisition, Phase-locked Loop (PLL),Frequency Division Multiplexing. Frequency Modulation Systems: Concept of Instantaneous Frequency, Generalized concept of Angle Modulation, Frequency modulation, Frequency Deviation, Spectrum of FM Signal with Sinusoidal Modulation, Bandwidth of FM Signal Narrowband and wideband FM, Bandwidth required for a Gaussian Modulated WBFM Signal, Generation of FM Signal, FM Demodulator, PLL, Pre-emphasis and De-emphasis Filters.
Module-III (12 Hours)
Mathematical Representation of Noise: Sources and Types of Noise, Frequency Domain Representation of Noise, Power Spectral Density, Spectral Components of Noise, Response of a Narrow band filter to noise, Effect of a Filter on the Power spectral density of noise, Superposition of Noise, Mixing involving noise, Linear Filtering, Noise Bandwidth, and Quadrature Components of noise. Noise in AM Systems: The AM Receiver, Super heterodyne Principle, Calculation of Signal Power and Noise Power in SSB-SC, DSB-SC and DSB, Figure of Merit ,Square law Demodulation, The Envelope Demodulation, Threshold
Module-IV (8 Hours)
Noise in FM System: Mathematical Representation of the operation of the limiter, Discriminator, Calculation of output SNR, comparison of FM and AM, SNR improvement using pre-emphasis, Multiplexing, Threshold infrequency modulation, The Phase locked Loop.
Text Books:
References Books:
Alternate form of Fourier Series is
The Fourier series hence expresses a periodic signal as an infinite summation of harmonics of
fundamental frequency (^0) 0
1 f T
= (^). The coefficients (^) C (^) n are called spectral amplitudes i.e. (^) C (^) n is the
amplitude of the spectral component 0
n cos n
nt C T
π φ
at frequency nf (^0). This form gives one sided
spectral representation of a signal as shown in1st^ plot of Figure 1.
Exponential Form of Fourier Series
This form of Fourier series expansion can be expressed as :
0
0 0 0
2 /
(^22) /
0 2
j nt T n n T j nt T n T
π
π
∞ =−∞
−
∑
∫
The spectral coefficients Vn and V-n have the property that they are complex conjugates of each other
V n = V (^) − n. This form gives two sided spectral representation of a signal as shown in 2 nd (^) plot of Figure-
0 0
n j n n
The Vn’s are the spectral amplitude of spectral components 0 j 2 ntf V en π .
0 (^1 ) 0 0 2 2
1
n n n
n n n n n n
π φ
φ
∞
=
−
The Sampling Function
The sampling function denoted as Sa(x) is defined as:
Sa x x
=
And a similar function Sinc(x) is defined as :
Sinc x x
π π
=
The Sa(x) is symmetrical about x=0 , and is maximum at this point Sa(x)=1. It oscillates with an
amplitude that decreases with increasing x. It crosses zero at equal intervals on x at every x^ =^ ± n^ π, where n is an non-zero integer.
Figure 2 Plot of Sinc(f)
Figure 1 One sided and corresponding two sided spectral amplitude plot
-3f 0 -2f 0 -f 0 0 f 0 2f 0 3f 0 frequencyÆ
0 f 0 2f 0 3f 0 frequencyÆ
Cn
Vn
4. Time Scaling Property 1 (at)
f x X a a
⎛ ⎞ ⇔ (^) ⎜ ⎟ ⎝ ⎠
5. Convolution Property: If convolution operation between two signals is defined as:
x (t) y(t) x (^) ( τ (^) ) ( x t τ (^) ) d τ
∞
−∞
6. Modulation Property (^20) (t) X(f f ) 0 e j^^ π^ f tx ⇔ − 7. Parseval’s Property
∞ ∞ ∗ ∗ −∞ −∞
8. Autocorrelation Property: If the time autocorrelation of signal x(t) is expressed as:
∞
−∞
2 Rx ( ) τ ⇔ X (f)
9. Differentiation Property: d (^) x (t) j 2 fX (f) dt
⇔ π
Response of a linear system
The reason what makes Trigonometric Fourier Series expansion so important is the unique characteristic of the sinusoidal waveform that such a signal always represent a particular frequency. When any linear system is excited by a sinusoidal signal, the response also is a sinusoidal signal of same frequency. In other words, a sinusoidal waveform preserves its wave-shape throughout a linear system. Hence the response-excitation relationship for a linear system can be characterised by, how the response amplitude is related to the excitation amplitude (amplitude ratio) and how the response phase is related to the excitation phase (phase difference) for a particular frequency. Let the input to a linear system be :
vi (^) ( t , ω n (^) )= V en j^^ ω nt
Then the filter output is related to this input by the Transfer Function (characteristic of the Linear
Filter): H (^) ( ω n ) = H (^) ( ω (^) n ) e − j^ θ^ (^ ω n ), such that the filter output is given as
vo ( t , ω n ) = Vn H ( ω n ) ej (^^ ω n^^ t^ − j θ ω(^ n ))
Normalised Power
While discussing communication systems, rather than the absolute power we are interested in another quantity called Normalised Mean Power. It is an average power normalised across a 1 ohm resistor, averaged over a single time-period for a periodic signal. In general irrespective of the fact, whether it is a periodic or non-periodic signal, average normalised power of a signal v(t) is expressed as :
( )
2 2
2
lim
1 T
T
T
P v t dt →∞ (^) T − = (^) ∫
Energy of signal
For a continuous-time signal, the energy of the signal is expressed as:
∞
−∞
A signal is called an Energy Signal if
0 0
A signal is called Power Signa l if
0 P E
Normalised Power of a Fourier Expansion
If a periodic signal can be expressed as a Fourier Series expansion as:
Then, its normalised average power is given by :
( )
2 2
2
lim 1 T
T
T
P v t dt →∞ (^) T − = (^) ∫
Integral of the cross-product terms become zero, since the integral of a product of orthogonal signals over period is zero. Hence the power expression becomes:
2 2 (^2 1 ) 0 ... 2 2
C C P = C + + +
By generalisation, normalised average power expression for entire Fourier Series becomes:
Expansion in Orthogonal Functions
Let there be a set of functions (^) g 1 (^) (x), g 2 (x), g (x), ..., g 3 n (x), defined over the interval (^) x 1 (^) < x < x 2 and
such that any two functions of the set have a special relation:
2
1
(x) g (x) dx 0
x i j x ∫ g^ =.
The set of functions showing the above property are said to be an orthogonal set of functions in the
sum of such (^) g (x) n ’s as:
f (x) = C g 1 1 (^) (x) + C (^) 2 g 2 (x) + C (^) 3 g (x) 3 + ... + Cn g (^) n (x), where^ C^ n ’s are the numerical coefficients
The numerical value of any coefficient C (^) n can be found out as:
2
1 2
1
2
(x) g (x)
(x) dx
x n x n x n x
f dx C g
=
In a special case when the functions (^) g (x) n in the set are chosen such that
2
1
(^2) (x) dx x n x
∫ g =1, then such a
set is called as a set of orthonormal functions , that is the functions are orthogonal to each other and each one is a normalised function too.
Amplitude Modulation Systems
In communication systems, we often need to design and analyse systems in which many independent message can be transmitted simultaneously through the same physical transmission channel. It is possible with a technique called frequency division multiplexing , in which each message is translated in frequency to occupy a different range of spectrum. This involves an auxiliary signal called carrier which determines the amount of frequency translation. It requires modulation, in which either the amplitude, frequency or phase of the carrier signal is varied as according to the instantaneous value of the message signal. The resulting signal then is called a modulated signal. When the amplitude of the carrier is changed as according to the instantaneous value of the message/baseband signal, it results in Amplitude Modulation. The systems implementing such modulation are called as Amplitude modulation systems.
Frequency Translation
Frequency translation involves translating the signal from one region in frequency to another region. A signal band-limited in frequency lying in the frequencies from f 1 to f^2 , after frequency translation can be translated to a new range of frequencies from f 1 ’^ to f 2 ’. The information in the original message signal at baseband frequencies can be recovered back even from the frequency-translated signal. The advantagesof frequency translation are as follows:
(t)max a
x m A
=
On the basis of modulation index, AM signal can be from any of these cases:
I. (^) m (^) a > 1 : Here the maximum amplitude of baseband signal exceeds maximum carrier
amplitude, x (t)^ max>^ A. In this case, the baseband signal is not preserved in the AM envelope, hence baseband signal recovered from the envelope will be distorted. II. (^) m (^) a ≤ 1 : Here the maximum amplitude of baseband signal is less than carrier amplitude x (t) (^) max≤ A. The baseband signal is preserved in the AM envelope.
Spectrum of Double-sideband with carrier (DSB+C)
Let x(t) be a bandlimited baseband signal with maximum frequency content f (^) m. Let this signal modulate a carrier c (t) = A C os(2 πf t) c .Then the expression for AM wave in time-domain is given by:
(t) (^) [ (t) (^) ] (2 f t) ACos(2 f t) x(t) Cos(2 f t)
c c c
s A x Cos π π π
Taking the Fourier transform of the two terms in the above expression will give us the spectrum of the DSB+C AM signal.
[ ]
[ ]
c c c
π δ δ
π
So, first transform pair points out two impulses at (^) f = ± fc , showing the presence of carrier signal in
the modulated waveform. Along with that, the second transform pair shows that the AM signal spectrum contains the spectrum of original baseband signal shifted in frequency in both negative and positive direction by amount (^) f (^) c. The portion of AM spectrum lying from (^) f (^) c to (^) f (^) c + fm in positive
frequency and from (^) − fc to (^) − f (^) c − fm in negative frequency represent the Upper Sideband(USB). The
portion of AM spectrum lying from (^) f (^) c − fm to (^) f (^) c in positive frequency and from (^) − f (^) c + fm to (^) − fc in
negative frequency represent the Lower Sideband(LSB). Total AM signal spectrum spans a frequency from (^) f (^) c − fm to (^) f (^) c + fm , hence has a bandwidth of (^2) f (^) m.
Power Content in AM Wave
By the general expression of AM wave:
s (t) = ACos(2 f t) π c +x(t)Cos(2 f t) π c
Hence, total average normalised power of an AM wave comprises of the carrier power corresponding to first term and sideband power corresponding to second term of the above expression.
/2 2 2 2 / / 2 2 2 /
total carrier sideband T carrier c T (^) T T sideband c T (^) T
π
π
∞
∞
→ (^) −
→ (^) −
In the case of single-tone modulating signal where x (t)^^ =^ V m^ Cos (2 f t)^ π m :
2
2
2 2
2
carrier
m sideband
m total carrier sideband
a total carrier
Where, ma is the modulation index given as (^) m (^) a Vm A
Net Modulation Index for Multi-tone Modulation : If modulating signal is a multitone signal expressed in the form:
x (t) = V 1 Cos (2 f t) π 1 +V 2 Cos (2 f t) π 2 +V 3 Cos (2 f t) ... π 3 + +V n (^) Cos (2 f t)π n
Then,
2 2 2 2 1 1 2 3 ... 2 2 2 2
n total carrier P = P ⎡^ + m^ + m + m^ m ⎤ ⎢ ⎥ ⎣ ⎦
Where (^) m 1 (^) V^1^^ , m (^) 2 V^^2 , m (^) 3 V^2 ,..., mn Vn A A A A
Generation of DSB+C AM by Square Law Modulation
Square law diode modulation makes use of non-linear current-voltage characteristics of diode. This method is suited for low voltage levels as the current-voltage characteristic of diode is highly non- linear in the low voltage region. So the diode is biased to operate in this non-linear region for this application. A DC battery Vc is connected across the diode to get such a operating point on the characteristic. When the carrier and modulating signal are applied at the input of diode, different frequency terms appear at the output of the diode. These when applied across a tuned circuit tuned to carrier frequency and a narrow bandwidth just to allow the two pass-bands, the output has the carrier and the sidebands only which is essentially the DSB+C AM signal.
Demodulation of DSB+C by Square Law Detector
It can be used to detect modulated signals of small magnitude, so that the operating point may be chosen in the non-linear portion of the V-I characteristic of diode. A DC supply voltage is used to get a fixed operating point in the non-linear region of diode characteristics. The output diode current is hence
Figure 6 Square Law Detector
given by the non-linear expression:
(t) b 2 (t) i = avFM + vFM
Where vFM^ (t)^ =[A^ +^ x (t)]Cos(2 f t)^ π c
This current will have terms at baseband frequencies as well as spectral components at higher frequencies. The low pass filter comprised of the RC circuit is designed to have cut-off frequency as the highest modulating frequency of the band limited baseband signal. It will allow only the baseband frequency range, so the output of the filter will be the demodulated baseband signal.
Linear Diode Detector or Envelope Detector
This is essentially just a half-wave rectifier which charges a capacitor to a voltage to the peak voltage of the incoming AM waveform. When the input wave's amplitude increases, the capacitor voltage is increased via the rectifying diode quickly, due a very small RC time-constant (negligible R ) of the charging path. When the input's amplitude falls, the capacitor voltage is reduced by being discharged by a ‘bleed’ resistor R which causes a considerable RC time constant in the discharge path making discharge process a slower one as compared to charging. The voltage across C does not fall appreciably during the small period of negative half-cycle, and by the time next positive half cycle appears. This cycle again charges the capacitor C to peak value of carrier voltage and thus this process repeats on. Hence the output voltage across capacitor C is a spiky envelope of the AM wave, which is same as the amplitude variation of the modulating signal.
Double S
If the car transmitte componen AM signa
s
So, the ex
s
Therefore signal. Th
Differenc still both t
bandwidth
Generatio
ideband Sup
rrier is suppr er power. Th nt of AM sign al is given by:
sD (^) SB C + (t) =AC
xpression for D
sD (^) SB SC − (t) =x(
e, a DSB-SC his is accompl
e from the the the sidebands h of (^2) f (^) m.
on of DSB-SC
ppressed Car
ressed and o his will not nal do not ca :
ACos(2 f t) π (^) c +
DSB-SC whe
x(t)Cos(2 f t π c
signal is obta lished by a pr
e DSB+C bei s, spectral spa
C Signal
Figure 7 E
rrier(DSB-SC
only the side affect the in arry any infor
+x(t)Cos(2 π
ere the carrier
t)
ained by sim roduct modu
Figure 8 P ing only the a an of this DSB
Envelope Detec
ebands are tr nformation c rmation about
πf t) c
r has been sup
mply multiplyi ulator or mixe
Product Modula absence of car BSC wave is
tor
ransmitted, th content of th t the baseban
ppressed can b
ing modulatin er.
ator rrier compone still (^) f (^) c − fm t
his will be a he AM signa nd signal varia
be given as:
ng signal x(t)
ent, and since to (^) f (^) c + fm , h
a way to sav al as the car ation. A DSB
) with the car
e DSBSC has hence has a
ving rrier B+C
rrier
Figure 9 Synchronous Detection of DSBSC
Let the received DSB-SC signal is :
r(t) =x(t)Cos(2 f t) π c
So after carrier multiplication, the resulting signal:
[ ]
2
c c c
c
c
π π π
π
π
The low-pass filter having cut-off frequency f (^) m will only allow the baseband term (^1) x(t) 2
, which is in the
pass-band of the filter and is the demodulated signal.
Single Sideband Suppressed Carrier (SSB-SC) Modulation
The lower and upper sidebands are uniquely related to each other by virtue of their symmetry about carrier frequency. If an amplitude and phase spectrum of either of the sidebands is known, the other sideband can be obtained from it. This means as far as the transmission of information is concerned, only one sideband is necessary. So bandwidth can be saved if only one of the sidebands is transmitted, and such a AM signal even without the carrier is called as Single Sideband Suppressed Carrier signal. It takes half as much bandwidth as DSB-SC or DSB+C modulation scheme.
For the case of single-tone baseband signal, the DSB-SC signal will have two sidebands :
The lower side-band: Cos (2 (f^^ π^ c −f )t)^ m^ =^ Cos (2 f t)^^ π m Cos^ (2 f t)^ π c^ + Sin (2 f t)Sin(2 f t)π^ m π c
And the upper side-band: Cos (2 (f^^ π^ c +f )t)^ m^ =^ Cos (2 f t)^^ π m Cos^ (2 f t)^ π c^ − Sin (2 f t)Sin(2 f t)π^ m π c
If any one of these sidebands is transmitted, it will be a SSB-SC waveform:
s (t) (^) SSB (^) = Cos (2 f t) π m Cos (2 f t) π c ± Sin (2 f t)Sin(2 f t)π (^) m π c
Where (+) sign represents for the lower sideband, and (-) sign stands for the upper sideband. The
modulating signal here is x (t)^^ =^ Cos (2 f t)^ π m , so let xh (t)^ =^ Sin(2 f t)^ π m be the Hilbert Transform
2
⎛ π⎞ ⎜ − ⎟ ⎝ ⎠
to a signal. So the expression
for SSB-SC signal can be written as:
s (t) (^) SSB (^) = x (t) Cos (2 f t) π c ± xh (t)Sin(2 f t) π c
Where xh^ (t)is a signal obtained by shifting the phase of every component present in (^) x (t)by 2
⎛ π⎞ ⎜ − ⎟ ⎝ ⎠
Generation of SSB-SC signal
Frequency Discrimination Method:
Figure 10 Frequency Discrimination Method of SSB‐SC Generation
The filter method of SSB generation produces double sideband suppressed carrier signals (using a balanced modulator), one of which is then filtered to leave USB or LSB. It uses two filters that have different passband centre frequencies for USB and LSB respectively. The resultant SSB signal is then mixed (heterodyned) to shift its frequency higher.
Limitations:
I. This method can be used with practical filters only if the baseband signal is restricted at its lower edge due to which the upper and lower sidebands do not overlap with each other. Hence it is used for speech signal communication where lowest spectral component is 70 Hz and it may be taken as 300 Hz without affecting the intelligibility of the speech signal. II. The design of band-pass filter becomes quite difficult if the carrier frequency is quite higher than the bandwidth of the baseband signal.
Phase-Shift Method:
