CT1037NCT1037NIntroduction to CommunicationsIntroduction to Communications

Signal RepresentationSignal Representation& Spectral Analysis& Spectral Analysis

Er. Saroj Sharan Regmi

Lecture 05

• Issues on Communication.

• Problems Affecting Communication.

• Antenna dipole lengths.

• Sinusoidal Carrier Modulation.

• Amplitude Modulation & Demodulation.

• Frequency Modulation & Demodulation.

• Multiplexing in Frequency, Wavelength or Time Division.

Last Lecture: 03 & 04Last Lecture: 03 & 04Analogue Modulation & MultiplexingAnalogue Modulation & Multiplexing

• Signals and Systems,

• Continuous- and Discrete- Time Systems,

• Continuous- and Discrete- Time Signals,

• Fourier Series,

• One-Sided Amplitude Frequency Spectrum,

• Two- Sided Amplitude Frequency Spectrum.

Today’s Lecture: 05Today’s Lecture: 05Signal Representation & Spectral AnalysisSignal Representation & Spectral Analysis

• Signal: Signal: a function of one or more variables that conveys information on the nature of a physical phenomenon.

• System: System: an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.

• System System analysisanalysis: : analyze the output signal when input signal and system are given.

• System synthesis: System synthesis: design the system when input and output signals are given.

Signals & SystemsSignals & Systems

Continuous-time System: Continuous-time System: the input and output signals are continuous in time.

Continuous Time SystemContinuous Time System

Discrete-time System: Discrete-time System: has discrete-time input and output signals.

Discrete Time SystemDiscrete Time System

Continuous-time SignalsContinuous-time Signals Discrete-time SignalsDiscrete-time Signals

Even Odd

Periodic Non-periodic/aperiodic

Deterministic Random

Energy Power

Signal ClassificationSignal Classification

• x(t): defined for all time t.

• x[n] : defined only at discrete instants of time.

• x[n] = x(nTs), n = 0, ±1, ±2, ±3, …

• Ts: sampling period

Continuous- and Discrete- Time SignalsContinuous- and Discrete- Time Signals

(a) Continuous-time signal

x(t).

(b) Representation of x(t)as a discrete-time signal x[n].

• Even Signal:Even Signal:

Symmetric about the vertical axis.

x(-t) = x(t) for all t.

Even and Odd SignalsEven and Odd Signals

• Odd Signal:Odd Signal:

Asymmetric about the vertical axis.

x(-t) = -x(t) for all t.

• Periodic Signal:Periodic Signal: x(t) = x(t+T), for all t

T = fundamental period

Fundamental frequency, f = 1/T (Hz)

Angular frequency, ω = 2πf (rad/s)

• Non-Periodic Signal:Non-Periodic Signal:

No value of T satisfies the condition above.

Periodic and Non-Periodic SignalsPeriodic and Non-Periodic Signals

• Periodic Signal:Periodic Signal:

Periodic and Non-Periodic Signals (…2)Periodic and Non-Periodic Signals (…2)

• Non-Periodic Signal:Non-Periodic Signal:

• What is the fundamental frequency of triangular wave below? Express the fundamental frequency in units of Hz and rad/s?

Periodic and Non-Periodic Signals (Example)Periodic and Non-Periodic Signals (Example)

T = 0.2 secs

f = 1/0.2 = 5 Hz

ω = 2πf = 2 x 3.14 x 5 = 31.42 rad/s

• Periodic discrete time signal: x[n] = x[n + N], for integer n

Periodic signal

Periodic and Non-Periodic SignalsPeriodic and Non-Periodic SignalsFor Discrete Time SignalsFor Discrete Time Signals

Non-Periodic signal

• Deterministic signal: there is no uncertainty with respect to its

value at any time. Specified function.

• Random signal: there is uncertainty before it occurs.

Deterministic and Random SignalsDeterministic and Random Signals

• Energy signal: 0 < E <

• Power signal: 0 < P <

Continuous Time Continuous Time SignalsSignals

Energy and Power SignalsEnergy and Power Signals

Discrete time signalsDiscrete time signals

Fourier SeriesFourier Series

• A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.

They make use of the orthogonality relationships of the sine and cosine functions.

• Most useful for a mathematical treatment of signals that are deterministic.

A deterministic waveform can be modelled as a completely specified function of time.

• For now, we shall only use periodic signals which are necessarily power signals, ie they have:

Finite average power,

Infinite energy.

A Periodic SignalA Periodic Signal

• An example of a periodic signal:

• The signal above can be represented by:

a series of sine and cosine terms,

plus a dc term (ie constant value, independent of frequency).

T 2Tt

x(t)

Frequency Component of A Periodic SignalFrequency Component of A Periodic Signal

• The Lowest frequency (other than dc) of the sinusoidal components is a frequency, f1, given by:

f 1 =1T

…(Hz )

• All other frequencies in the signal will be integer multiples of the fundamental frequency, f1, and these are called harmonics.

• Thus:

f1 is the fundamental frequency or 1st harmonic,

f2 is the 2nd harmonic, even,

f3 is the 3rd harmonic, odd,

f4 is the 4th harmonic, even,

fn is the nth harmonic.

The Periodic SignalThe Periodic Signal

f1

f2

f3

The resultant periodic signal:(f1 + f2 + f3)

Bandwidth IssuesBandwidth Issues

• In any transmission system the bandwidth must be sufficiently large to pass all significant frequencies on the signal.

• Mathematically, signals contain an infinite number of harmonics suggesting that transmission systems must have infinite bandwidth.

• In reality this is not so as there are always significant frequencies that can be used to efficiently reproduce the signal, ie some frequency components, usually the weaker ones, can be omitted from the signal and still be able to recognise the signal without noticeable degradation of its quality.

Expressing SignalsExpressing Signals

• Any periodic signal may be expressed as the sum of sine and/or cosine terms:

This is the ‘sine-cosine’ fourier form.

• Mathematically:

x ( t ) = A0 + ∑n=1

n=∞

( Ancos nω1 t + Bnsin nω1 t )

= A0 + A1 cos nω1 t + B1 sin nω1 t+ A2 cos nω1 t + B2 sin nω1 t+ A3 cos nω1 t + B3 sin nω1 t

+ ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯+ An cosnω1 t + Bnsin nω1 t

ω = 2πfThus :nω1 t =2 πnf 1 t2 πntT

Note:

Harmonic Signal StrengthHarmonic Signal Strength

• The signal strength for a particular harmonic is given by:

C n = √ A n2 + Bn2

ie, for the second harmonic term:

C 2 = √ A 22 + B 2

2

Note: often the signal may contain only A or B coefficient terms.

One-Sided Amplitude Frequency SpectrumOne-Sided Amplitude Frequency Spectrum

• Shows the amplitudes C versus frequency.

• It is not important to show the phase in these plots.

• Loosely referred to as the ‘Frequency Spectrum’.

0 f1 f2 f3 f (Hz)

C1 C2

C3

C0

Cn

Two-Sided Amplitude Frequency SpectrumTwo-Sided Amplitude Frequency Spectrum

• Results from the complex representation of the Fourier Series:

where we also consider that we have both positive and negative frequencies.

x ( t ) = ∑n=−∞

n=0

X n ejnωi t

It can be shown that the magnitude of is given by:X n

|X n|= X n = √ A n2+B n24for n≠0

Two-Sided Amplitude Frequency Spectrum (…2)Two-Sided Amplitude Frequency Spectrum (…2)

• Hence:X n =

Cn2

with X0 = C0

• Thus:

X n =Cn2

C1

2C2

2 C3

2

C1

2C2

2C3

2

X 0 = C 0 = A 0

0 f1 f2 f3-f3 -f2 -f1f (Hz)

SummarySummary

• Signals and Systems,

• Continuous- and Discrete- Time Systems,

• Continuous- and Discrete- Time Signals,

• Fourier Series,

• One-Sided Amplitude Frequency Spectrum,

• Two- Sided Amplitude Frequency Spectrum.