1. SIGNALS & SYSTEMS

2. Continuous-time signal
A signal x(t) is a continuous-time signal if t is a continuous variable
Discrete time signal
If ‘t’ is a discrete variable, that is, x(t) is defined at discrete times, then x(t) is a discrete-time signal.

3. SIGNAL
Signal is a physical quantity that varies with respect to time , space or any other independent variable
Eg x(t)= sin t.
the major classifications of the signal are:
(i) Discrete time signal
(ii) Continuous time signal

4. Unit Step &Unit Impulse
Discrete time Unit impulse is defined as
δ [n]= {0, n≠ 0
{1, n=0
Unit impulse is also known as unit sample.
Discrete time unit step signal is defined by
U[n]={0,n=0
{1,n>= 0
Continuous time unit impulse is defined as
δ (t)={1, t=0
{0, t ≠ 0
Continuous time Unit step signal is defined as
U(t)={0, t<0
{1, t≥0

5. SIGNAL A signal is said to be periodic ,if it exhibits periodicity.
Periodic Signal & Aperiodic Signal
A signal is said to be periodic ,if it exhibits periodicity.
i.e., X(t +T)=x(t), for all values of t. Periodic signal has the property that it is unchanged by a time shift of T. A signal that does not satisfy the above periodicity property is called an aperiodic signal
even and odd signal ?
A discrete time signal is said to be even when, x[-n]=x[n]. The continuous time signal is said to be even when, x(-t)= x(t) For example, Cosωn is an even signal.

6. Energy and power signal
A signal is said to be energy signal if it have finite energy and zero power.
A signal is said to be power signal if it have infinite energy and finite power.
If the above two conditions are not satisfied then the signal is said to be neither energy nor power signal

7. SYSTEMS AND CLASSIFICATION OF SYSTEMS
A system is an operation that transforms input signal x into output signal y. SYSTEM REPRESENTATION

8. CONTINUOUS-TIME AND DISCRETE-TIME SYSTEMS

9. SYSTEMS WITH MEMORY AND WITHOUT MEMORY
A system is said to be memory less if the output at any time depends on only the input at that same time. Otherwise, the system is said to have memory
y(t) = Rx(t)
An example of a system with memory is a capacitor C with the current as the input x( t ) and the voltage as the output y ( t) ; then 
y(t) = 1/c ʃ x(τ) dτ
A second example of a system with memory is a discrete-time system whose input and output sequences are related by 
y[n] = Σ x[n]
CAUSAL AND NONCAUSAL SYSTEMS
y(t) = x(t+1)
y[n] = x[-n]

10. LINEAR SYSTEMS AND NONLINEAR SYSTEMS
.Additivity: Given that Tx1 = y1 and Tx2 = y2, then T{x1 +x2} =y1 +y2 for any signals x1 and x2. 2. Homogeneity (or Scaling): T{αx} = αy for any signals x and any scalar α. Nonlinear system. Equations (1.40) and ( 1.41) can be combined into a single condition as T { α1x1+α2x2} = α1y1+α2y2 where α1 and α2 are arbitrary scalars. Examples of linear systems are the resistor and the capacitor. Examples of nonlinear systems are y = x² y = cos x

11. Time-Invariant and Time-Varying Systems
{x(t- τ)} = y(t-τ) T{ x [n-k]} =y[n-k]

12. FEEDBACK SYSTEMS
A special class of systems of great importance consists of systems having feedback.

13. LINEAR TIME INVARIANT SYSTEMS
Two most important attributes of systems are linearity and time-invariance. The input-output relationship for LTI systems is described
in terms of a convolution operation. The importance of the convolution operation in LTI
systems stem from the fact that knowledge of the response of an LTI system to the unit
impulse input allows us to find its output to any input signals

14. RESPONSE OF A CONTINUOUS-TIME LTI SYSTEM
IMPULSE RESPONSE
The impulse response h(t) of a continuous-time LTI system (represented by T) is
Defined to be the response of the system when the input is δ(t), that is, h(t) = T{ δ(t)}
RESPONSE TO AN ARBITRARY INPUT:
input x( t) can be expressed as 
x(t) = x(τ) δ(t- τ)dτ
Since the system is linear, the response y(t) of the system to an arbitrary input x( t ) can be expressed as
y(t) = T{x(t)} = T{x(τ) δ(t- τ)d
= x(τ) δ(t- τ)dτ
Since the system is time-invariant, we have
h(t-τ) = T{δ(t-τ) } 
y(t) = x(τ) h(t- τ)dτ 

15. Fourier Series
:The Fourier series represents a periodic signal in terms of frequency components
We get the Fourier series coefficients as follows:
The complex exponential Fourier coefficients are a sequence of complex numbers representing the frequency component ω0k.

16. Fourier series
Fourier series: a complicated waveform analyzed into a number of harmonically related sine and cosine functions
A continuous periodic signal x(t) with a period T0 may be represented by:
X(t)=Σ∞k=1 (Ak cos kω t + Bk sin kω t)+ A0
Dirichlet conditions must be placed on x(t) for the series to be valid: the integral of the magnitude of x(t) over a complete period must be finite, and the signal can only have a finite number of discontinuities in any finite interval

17. Laplace Transform
Lapalce transform is a generalization of the Fourier transform in the sense that it allows “complex frequency” whereas Fourier analysis can only handle “real frequency”. Like Fourier transform, Lapalce transform allows us to analyze a “linear circuit” problem, no matter how complicated the circuit is, in the frequency domain in stead of in he time domain.
Mathematically, it produces the benefit of converting a set of differential equations into a corresponding set of algebraic equations, which are much easier to solve. Physically, it produces more insight of the circuit and allows us to know the bandwidth, phase, and transfer characteristics important for circuit analysis and design.
Most importantly, Laplace transform lifts the limit of Fourier analysis to allow us to find both the steady-state and “transient” responses of a linear circuit. Using Fourier transform, one can only deal with he steady state behavior (i.e. circuit response under indefinite sinusoidal excitation).
Using Laplace transform, one can find the response under any types of excitation (e.g. switching on and off at any given time(s), sinusoidal, impulse, square wave excitations, etc.

18. Laplace Transform

19. Application of Laplace Transform to Circuit Analysis

20. PROPERTIES OF THE LAPLACE TRANSFORM
Basic properties of the Laplace transform are presented in the following.
Linearity: If
x1(t)< > X1(s) ROC = R1
x2(t)< > X2(s) ROC = R2
a1x1(t)+a2x2(t) < > a1X1(s)+a2X2(s)

21. TIME SHIFTING x(t)<------> X(S) ROC = R then
x(t-to)< > exp(-sto)X(s)  
SHIFTING IN THE S-DOMAIN:
Then ,x(t)< > X(s-so)  
TIME SCALING:
x(t)< > X(s) ROC=R
x(at)< >1/|a| X(s/a) .
TIME REVERSAL:
x(t) < >X(s) ROC = R
x(-t) < >X(-s)
Thus, time reversal of x(t) produces a reversal of both the σ and jω-axes in the s-plane.

22. DIFFERENTIATION IN THE TIME DOMAIN:
x( t)< >X(S) ROC = R
then ,dx(t)/dt< > sX(s)  
DIFFERENTIATION IN THE S-DOMAIN:
x(t)< > X(s) ROC=R
-tx(t)< >dX(s)/ds
INTEGRATION IN THE TIME DOMAIN
x(t)< >X(s) ROC=R
Then, x(τ)dτ< >1/s X(s)  
CONVOLUTION:
x1(t)< >X1(s) ROC=R1
x2(t)< >X2(s) ROC=R2
Then ,x1(t)* x2(t)< > X1(s) X2(s) 

23. Time-Invariance &Causality
If you delay the input, response is just a delayed version of original response.
X(n-k) y(n-k)
Causality could also be loosely defined by “there is no output signal as long as there is no input signal” or “output at current time does not depend on future values of the input”.

24. Convolution
The input and output signals for LTI systems have special relationship in terms of convolution sum and integrals.
Y(t)=x(t)*h(t) Y[n]=x[n]*h[n]

25. Sampling theory
The theory of taking discrete sample values (grid of color pixels) from functions defined over continuous domains (incident radiance defined over the film plane) and then using those samples to reconstruct new functions that are similar to the original (reconstruction).
Sampler: selects sample points on the image plane
Filter: blends multiple samples together

26. Sampling theory
For band limited function, we can just increase the sampling rate
• However, few of interesting functions in computer graphics are band limited, in particular, functions with discontinuities.
• It is because the discontinuity always falls between two samples and the samples provides no information of the discontinuity.

27. Sampling theory

28. Aliasing

29. Z-transforms
For discrete-time systems, z-transforms play the same role of Laplace transforms do in continuous-time systems
As with the Laplace transform, we compute forward and inverse z-transforms by use of transforms pairs and properties
Bilateral Forward z-transform
Bilateral Inverse z-transform

30. PROPERTIES OF THE Z-TRANSFORM

31. Region of Convergence
Region of the complex z-plane for which forward z-transform converges
Four possibilities (z=0 is a special case and may or may not be included)
Im{z}
Re{z}
Entire plane
Im{z}
Re{z}
Disk
Im{z}
Re{z}
Complement of a disk
Im{z}
Re{z}
Intersection of a disk and complement of a disk

32. Z-transform Pairs h[n] = an u[n] h[n] = d[n] h[n] = d[n-1]
Region of convergence: |z| > |a| which is the complement of a disk
h[n] = d[n]
Region of convergence: entire z-plane
h[n] = d[n-1]
h[n-1]  z-1 H[z]

33. Inverse z-transform

34. DISCRETE FOURIER TRANSFORM (DFT):
1. X(T) --- CONTINUOUS-TIME SIGNAL
X(f) --- Fourier Transform, frequency characteristics
if we don’t have a mathematical equation for x(t
(1) Sample x(t) =>
x0, x1, … , xN-1 over T (for example 1000 seconds)
Sampling period (interval)
N (samples) over T =>
Can we have infinite T and N? Impossible!

35. References:
Signals and systems-Simon haykin and Van Veen, Wiley. 2nd Edition.
Fundamentals of signals and system Michel J. Robert, MGH International Edition, 2008.

36. ANALOG COMMUNICATION

37. Impressing it on a carrier signal of the Form.
Amplitude modulation
Amplitude modulation is a type of modulation where the amplitude of the carrier signal is varied in accordance with the information bearing signal.
The message signal m(t) is transmitted through the communication channel by
Impressing it on a carrier signal of the Form.
c(t) = Ac cos(2πfct + φc)

38. Different methods of Amplitude modulation
Double side band suppressed carrier AM
Conventional double sideband AM
Single sideband AM
Vestigial sideband AM

39. Double sideband suppressed carrier
Double-sideband suppressed-carrier transmission (DSB-SC):  transmission in which frequencies produced by amplitude modulation are symmetrically spaced above and below the carrier frequency and the carrier level is reduced to the lowest practical level, ideally completely suppressed.

40. Conventional Amplitude Modulation
A conventional AM signal consists of a large carrier component in addition to the Double-sideband AM modulated signal. The transmitted signal is expressed mathematically as
u(t) = Ac[l +m(t)] cos(2πfc t +¢c)

41. SINGLE-SIDEBAND AMPLITUDE MODULATION
the carrier and one of the sidebands of an amplitude-modulated waveform are suppressed. This saves on bandwidth occupancy and signal power.
SSB-AM signal can be generated using Hilbert transform.

42. VESTIGIAL-SIDEBAND AM
Vestigial sideband (VSB) is a type of amplitude modulation technique (sometimes called VSB-AM ) that encodes data by varying the amplitude of a single carrier frequency . Portions of one of the redundant sidebands are removed to form a vestigial sideband signal so called because a vestige of the sideband remains.
In the time domain the VSB signal may be expressed as

43. Generation of AM signals
Balanced modulator
Ring modulator
Envelope detector

44. Balanced modulator
A relatively simple method to generate a DSB-SC AM signal is to use two conventional AM modulators
 A modulator constructed so that the carrier is suppressed and any associated carrier noise is balanced out. 

45. Ring modulator This is also for generating a DSB-SC AM signal.
Ring modulation is a signal-processing effect in electronics, an implementation of amplitude modulation or frequency mixing, performed by multiplying two signals, where one is typically a sine wave or another simple waveform.

46. Envelope detector
conventional DSB AM signals are easily demodulated by means of an envelope detector.
An envelope detector is an electronic circuit that takes a high-frequency signal as input and provides an output which is the "envelope" of the original signal.
Equation of AM can be written as. Where R(t) is envelope of the signal

47. Signal multiplexing
In telecommunication  and  computer networks, multiplexing (also known as muxing) is a process where multiple analog message signals or digital data streams are combined into one signal over a shared medium.
There are 2 most commonly used methords of multiplexing are
Frequency division multiplexing
Time division multiplexing

48. Frequency division multiplexing
Frequency-division multiplexing (FDM) is inherently an analog technology .
FDM achieves the combining of several digital signals into one medium by sending signals in several distinct frequency ranges over that medium.

49. Time division multiplexing
Time division multiplexing (TDM) is a digital technology. 
Time Division Multiplexing is the process of dividing up one communication time slot into smaller time slots.

50. Angle modulation
 These techniques are based on altering the angle (or phase) of a sinusoidal carrier wave to transmit data.
Angle Modulation is modulation in which the angle of a sine-wave carrier is varied by a modulating wave.
There are 2 types of angle modulation
Frequency modulation
Phase modulation

51. Frequency modulation
In telecommunication and signal processing, frequency modulation (FM) conveys information over a carrier wave by varying its instantaneous  frequency.
Frequency modulation uses the information signal, Vm(t) to vary the carrier frequency within some small range about its original value. Here are the three signals in mathematical form:
Information: Vm(t)
Carrier: Vc(t) = Vco sin ( 2 p fc t + f )
FM: VFM (t) = Vco sin (2 p [fc + (Df/Vmo) Vm (t) ] t + f)
Simple FM wave

52. Applications of frequency modulation
Magnetic tape storage
Sound
Radio

53. Magnetic tape storage
FM is also used at intermediate frequencies by all analog VCR systems, to record both the  luminance (black and white) and the chrominance portions of the video signal.
FM is the only feasible method of recording video to and retrieving video from magnetic tape without extreme distortion, as video signals have a very large range of frequency components — from a few hertz  to several megahertz. 

54. Sound FM is also used at audio frequencies to synthesize sound.
Signals is varied in both FM and AM

55. Radio
Frequency modulation (FM) is widely applied in Amateur Radio for voice, telegraphy and data modulation.
FM is commonly used at VHf radio frequencies for high fidelity broadcasts of music and speech (see FM broadcasting).

56. Phase modulation
Phase modulation (PM) is a form of modulation that represents information as variations in the instantaneous phase of a carrier wave.
In phase modulation, the instantaneous phase of a carrier wave is varied from its reference value by an amount proportional to the instantaneous amplitude of the modulating signal

57. References
Principles of Communication System – Simon haykin, John Wiley, 2nd Edition
Communication System Second Edition –R.P. Singh, SP Sapre, TMH, 2007

58. CELLULAR & MOBILE COMMUNICATIONS

59. Cellular Concept
Idea: replace high power transmitter with several lower power transmitters to create small “cells”
Multiple cells cover a geographic area
Each cell assigned a set of frequencies
Neighboring cells assigned different group of frequencies to reduce adjacent-cell interference

60. Cellular Communication System
It provide wireless connection from users to PSTN or between its users.
Use “cells” in order to increase the total capacity, given a limited spectrum, by re-using the frequency over different areas.
Use a Handover mechanism to enable an uninterrupted call connection when users move from one cell to another.

61. CELLULAR MOBILE COMMUNICATION

62. Level which HO occurred
Handover Mechanism
Level at B
RSL
Level which HO occurred
time
BS1
BS2 A B

63. GSM Architecture VLR HLR GMSC PSTN A interface Um interface CCITT
Base Transceiver
Station (BTS)
Base Station
Controller (BSC)
Abis interface
Base Station (BS)
Mobile Stations
(MS)
Um interface
A interface
CCITT
Signalling
System No. 7
(SS7)
interface
Mobile
Switching
Centre
(MSC)
GMSC
PSTN
VLR
HLR

64. Mobile Station MS consist of : Mobile Equipment (ME)
Subscriber Identification Module (SIM)

65. Base Transceiver Station (BTS)
BSS consist of two part :
Base Transceiver Station (BTS)
Base Station Controller (BSC)
BTS is a radio-end which determine a cell coverage and provide link with MS.
BTS include Transmitters and Receivers, antenna and signal processing unit as well as interface.
BTS communicate with MS via Um (air) interface

66. Base Station Controller
BSC control RRM for BTSs.
BSC handle radio-channel setup, frequency hopping, and handover within BSC

67. REFERENCES
J. E. Flood. Telecommunication Networks. Institution of Electrical Engineers, London, UK, 1997.s
Bernhard H. Walke. Mobile Radio Networks: Networking, protocols and traffic performance. John Wiley and Sons, LTD West Sussex England, Chapter 2.