The document covers various types of signals and systems, detailing their characteristics and classifications such as continuous, discrete, deterministic, and non-deterministic signals. It discusses mathematical concepts such as convolution, Fourier series, Laplace transforms, and z-transformation, along with their properties and applications in signal processing. Additionally, it explains the principles of modulation, demodulation, and multiplexing in communication systems.

2. SIGNAL:
Signal is a function of a single independent variable and it
contains some information.Amplitude varies with respect to independent
variable. TYPES OF SIGNALS:
1.Continuous and Discrete time signals
2.Deterministic and Non-deterministic signals
3.Causal and Non-causal signals
4.Bounded and Unbounded signals
5.Even and Odd signals
6.Periodic and Aperiodic signals
7.Energy and Power signals
SOME BASIC SIGNALS:
1.Unit Step function
2.Impulse Function
3.Unit Parabole
4.Rectangular Pulse
5.Triangular Pulse
Operations on Signals:
1.Time shifting
2.Time Scaling
3.Amplitude shifting
4.Amplitude Scaling

3. SYSTEMS:
An entity which processes signals and produces new signals is called
System.
TYPES OF SYSTEMS:
1.Static and Dynamic Systems
2.Causal and Non-Causal Systems
3.Time variant and Time Invariant Systems
4.Linear and Non-Linear Systems
5.Stable and Unstable Systems
6.Feedback and Non-Feedback Systems
7.Invertible and Non-Invertible Systems

4. LINEAR AND TIME INVARIANT SYSTEMS:
Linear and Time Invariant Systems are the combination of linearity and
Time invariance.LTI systems satisfies the superposition principle.

5. CONVOLUTION:
IMPULSE RESPONSE:
Impulse response is the output of the system when the impulse signal is
and it is denoted by h(t).
CONVOLUTION INTEGRAL CONVOLUTION SUM

6. CONVOLUTION METHODS:
1.Graphical Convolution
2.Analytical Method
3.Tabular Convolution
PROPERTIES OF CONVOLUTION:
1.Commutative property:y=x(t)*h(t)=h(t)*x(t);
2.Associative Property:y(t)=x(t)*{h1(t)*h2(t)}={x(t)*h1(t)}*h2(t)
3.Distributive Property:y(t)=x(t)*{h1(t)+h2(t)}=x(t)*h1(t)+x(t)*h2(t)
4.Time Invariance Property:x(t-t0)*h(t-t1)=y[t-(t0+t1)]
NOTE:
u(t)*u(t)=r(t)
⇒
x(t)*del(t)=x(t)
⇒
u(n)*u(n)=r(n+1)
⇒

7. FOURIER SERIES:
Consider a periodic signal x(t) with a fundamental period T,i.e.
x(t+T)=x(t) t, f
∀ 0=1/T;f0=fundamental frequency.
A periodic signal x(t) can be expressed as linear combination of sinusoids with
discrete frequencies which are multiples of f0.
COMPLEX FOURIER SERIES
TRIGONOMETRY FOURIER SERIES

8. COMPACT FROM OF FOURIER
SERIES :
RELATIONSHIP BETWEEN DIFFERENT
FOURIER SERIES
COEFFICIENTS:

9. DISADVANTAGE OF FOURIER SERIES:
Fourier series is only restricted to periodic signals,we can not construct aperiodic
using Fourier Series.So we assume that aperiodic signal as periodic signal with
infinite period.
x(t)
xT(t)

10. Fourier series of xT(t):
Consider a continuous function of w
As T0→ ∞, ω0becomes infinitesimal (ω0 → 0).

11. FOURIER TRANSFORM:
t=continuous time index in seconds
w=continuous frequency index in radians per
second
INVERSE FOURIER TRANSFORM:
FORMULA:
FORMULA:

12. DIRICHLET CONDITIONS:
If a signal satisfies the dirichlet conditions,
fourier transform for that signal will exist.Dirichlet condition are sufficient
conditions only not neccesary conditions
1.x(t) is absolutely integrable
2.x(t) has only a finite number of extrema in any finite interval
3.x(t) has only finite number of discontinuties in any finite interval
PARSAVAL’S RELATION:
If x(t) and X(w) are fourier transform pair,then

13. FOURIER TRANSFORM OF A PERIODIC SIGNAL:
Consider the Fourier Transform pair,
By the property of translation in frequency domain
Consider the x(t) with the fundamental period T

14. By the linearity property of Fourier Transform,
Therefore,
Therefore the F.T of x(t) is impulse train occuring at integer multiples
of 1/T with strength ck

15. LAPLACE TRANSFORM:
For a signal x(t),laplace transform X(s)
Is defined by
The signal x(t) is said to be the inverse Laplace transform of X(s),if
The constant c is choosen so that the lapalce transform X(s)
converges.The signal x(t) is said to be the inverse Laplace
transform of X(s),Symbolically
and
s +jw
=�

16. REGION OF CONVERGENCE:
The range of values of for which the X(s) integral converges
�
is know as the region of convergence.
Properties of ROC:
1.ROC is the vertical line parallel to the jw-axis.
2.If x(t) is a right sided signal,ROC is the right of the rightmost
pole >pole(max)
.�
3.If x(t) is a left sided signal, ROC is the left of the leftmost pole <pole(min)
.�
4.No poles in ROC
5.if x(t) is bilateral signal ROC is in between two poles
6.If ROC includes jw axis,then Fourier Transform of the x(t) exists—Stable
7.If ROC is greater than pole,then it is causal signal
8.If x(t) is finite duration signal,it’s ROC is entire plane

17. INITIAL VALUE THEOREM:
The initial value theorem states that,
FINAL VALUE THEOREM:
The Final value theorem states that,

18. SAMPLING THEOREM
DEFINITION:
Any band limited signal limited to |w|<wn can be
completely reconstructed from it’s samples if the samples are
taken at the rate of fs≥2fm .
↦ Nyquist sampling rate = 2fm
There are three types of sampling
1. Ideal Sampling
2. Natural sampling
3. Flat -Top Sampling

19. IDEAL SAMPLING: (SAMPLING WITH IMPULSES)

20. NATURAL SAMPLING:(SAMPLING WITH PULSES)

21. FLAT-TOP SAMPLING:(TAKE SAMPLE AND HOLD)
APERTURE EFFECT
Increase in pulse width in fat top sampling leads to attenuationof high frequencies in
reproduction , this is known as “Aperture Efect”

22. DISCRETE TIME FOURIER SERIES:(DTFS)
Representation of discrete time sequence in terms of complex
exponentials.
Equations
Discrete Fourier representation of a Sequence x[n] is given by
where
N=length of the discrete time sequence and
PARSEVAL
’S POWER
THEOREM

23. DISCRETE TIME FOURIER TRANSFORM:(DTFT)
Frequency domain analysis of Discrete time signals.
Equations
DTFT of a Sequence x[n]
IDTFT of is
Observe that Is continuous function of ⍵ and periodic with period 2π
exists only if

24. DISADVANTAGE OF DTFT:
The drawback in DTFT is that the frequency domain
representation of a discrete time signal obtained from DTFT will be a
continuous function of ⍵ and so it can’t be processed by digital system.
To Overcome this DFT is developed from DTFT by sampling ⍵ by a
finite number of samples.
DEVELOPMENT OF DFT FROM DTFT
Put In DTFT to get DFT of x[n] Where k=0,1,. . . ,N-1
N= Number of samples
Generally DFT with N number of Samples is known as N-Point
DFT

25. DISCRETE FOURIER TRANSFORM:(DFT)
DEFINITION OF DFT
Let x[n] be the discrete time signal of Length ‘L’
X(K) is N-Point DFT OF x(n) where N≥L is defined as
….
DEFINITION OF IDFT
Inverse DFT of X(k) is x[n] and is defined as…

26. TWIDDLE FACTOR OR PHASE FACTOR:
Twiddle factor is defined as
PROPERTIES OF TWIDDLE FACTOR
1
.
2
3

27. MATRIX METHOD FOR COMPUTATION OF DFT:
Finding DFT using Normal procedure may be lengthy so, we will go for this matrix method
In general the N-Point DFT matrix can be represented
as
where

28. MATRIX MODEL FOR 2- POINT DFT:
MATRIX MODEL FOR 4- POINT DFT:

29. SOME IMPORTANT PROPERTIES OF DFT:
1.Periodicity : X(k+N)=X(k)
2. Symmetry of Real
Sequence:
If x[n] is real then X*
(k)=X(N-
k)
3. Interpolation in Time
domain:
4.DFT{DFT{x[n]}} = Nx[N-n]

30. Z TRANSFORMATION:
Z Transform is digital equivalent of Laplace Transform
Defnition
The Z transform of discrete signal x[n] is defined as…..
The Inverse Z transform is defined as …..
ROC: Region of convergence
The region of convergence (ROC) is the set of points in the complex plane for which
the Z-transform summation converges.
ROC of the Z Transform is the circular disk centered at the
origin.

31. SOME PROPERTIES OF ROC OF Z TRANSFORM:
1. ROC of z-transform is indicated with circle in z-plane.
2. ROC does not contain any poles.
3. If x(n) is a finite duration causal sequence or right sided sequence, then the ROC is entire z-
plane except at z = 0.
4. If x(n) is a finite duration anti-causal sequence or left sided sequence, then the ROC is entire z-
plane except at z = ∞.
5. If x(n) is a infinite duration causal sequence, ROC is exterior of the circle with radius a. i.e. |z|
> a.
6. If x(n) is a infinite duration anti-causal sequence, ROC is interior of the circle with radius a. i.e.
|z| < a.
7. If x(n) is a finite duration two sided sequence, then the ROC is entire z-plane except at z = 0 &
z = ∞.
8. STABILITY : If ROC includes unit circle then it is stable.
RELATION BETWEEN DTFT , Z TRANSFORMS:
Put In Z transform expansion then we can have
DTFT

32. FILTERS:
5 types of filters
1.Low Pass
FIlter
2.High Pass Filter
3.Band Pass Filter 4.Band Reject /Notch
Filter

33. Allows all frequencies.
Poles and zeros of all pass filter are
reciprocal conjugates i.e.
,

34. COMMUNICATION SYSTEMS

35. Block Diagram of Analog Communication:

36. Random Process:
The sample space or ensemble composed of functions of time is called
“random process or stochastic process” denoted as X(t,s).
Stationary:
A random process is said to be “stationary” if all its statistical properties
(ex:mean ,variance, moments etc) do not change with time.
Wide Sense Stationary(WSS)
Strict Sense Stationary(SSS)
All SSS processes are WSS but not all WSS processes are SSS.

37. Ergodic process:
Correlation: 1.Autocorrelation 2.Crosscorrelation
Covariance:
The independent processes are uncorrelated.
The converse case is not necessarily true.
Gaussian Process:
Gaussian Random Variable:

38. If the samples of a random process are jointly Gaussian Random
Variables, then the process is Gaussian process.
WSS means SSS ,it is true for Gaussian process only.

39. Center limit theorem:
When independent random variables are added, their properly normalized
sum tends towards a normal or Gaussian distribution.

40. Power spectral density:
It is how the power is distributed through the frequency.
Power spectral density and autocorrelation are fourier transform pairs.
White Noise:
The spectral density is constant for all frequencies.

41. White noise is a wideband signal.
Modulation
1.Baseband signal 2.Efficient Radiation
3.Avoid interference 4.Noise reduction
Primary communication resources:
1.Bandwidth 2.Power
Hilbert Transform:
The phase angles of all components of a given signal are shifted by +90
or -90 degrees, then resulting function of time is known as “Hilbert
Transform” of the signal.
Application: SSB(single sideband)

42. Continous Wave Modulation:
Modulation is a process by which some of the characteristics of the carrier
is varied in accordance with message signal.
Terms: message signal, modulating signal, modulated signal
In this there are 2 types
1.Amplitude modulation 2.Angle modulation
Amplitude modulation:
1.DSBFC
S(t)=Ac[1+ka*m(t)]cos(2*pi*fc*t)
Types:under modulation, critical modulation, over modulation.
Single tone and Multi tone modulation:

44. Modulation Efficiency=33.33%
Carrier Power=66.67%
Generation of AM waves: 1.Square Law modulator
2. Switching modulator
Demodulation of AM waves: 1.Envelope detector
2. Square law demodulator 3.Coherent demodulation
2. DSBSC
S(t)=Ac*m(t)*cos(2*pi*fc*t)
Modulation efficiency=100%
Generation of DSBSC waves: 1. Nonlinear modulator 2. Switching
modulators (Diode bridge modulator and ring modulator) 3. Balanced
modulator

46. Demodulation of DSBSC waves:
1.Coherent detection
Effect of loss of coherence: 1. Quadrature Null effect (zero demodulated
signal) 2. Distorted signal
3. SSBSC
Generation of SSBSC waves: Phase discrimination method
Modulation efficiency=100%
Bandwidth=fm (message bandwidth)
Ideal filters do not exist.

47. 4.VSBSC

48. Angle Modulation:
Frequency Modulation:
Generation of FM using PM:

50. Phase Modulation:
Generation of PM using FM:

52. Multiplexing:
Multiplexing is a technique where by a number of independent signals can
be combined into a composite signal suitable for transmission over a
common channel.
1. Quadrature carrier multiplexing(QCM) or Quadrature amplitude
modulation(QAM)
2. Frequency Division Multiplexing(FDM)
Quantization:
The conversion of an analog signal into digital signal is called Quantization.

53. Midriser quantizer: L=2^n
Midtread quantizer: L=(2^n)-1
where L=number of representation levels
n=number of bits/sample
For every increase in single bit ,(SNR)o in dB increases by 6 dB.
Non Uniform Quantization:
LLOYOD MAX Quantizer:
Companding: 1. U law companding 2. A-law companding

54. Pulse Code Modulation:
Differential Pulse Code Modulation:

55. Delta Modulation:

56. Quantization noises:
1.Slope overload distortion 2.granular noise
Adaptive Delta Modulation:
Step size is adopted to the level of input signal.

57. Time-Division Multiplexing:
Several signals are multiplexed in the time domain to form a composite
signal for transmission over a channel.

58. Figure of Merit(FOM):
FOM=(SNR)o/(SNR)c
DSBSC modulation with coherent reception:
FOM=1
SSBSC modulation with coherent reception:
FOM=1
AM receivers using envelope detection:
FOM<1