Chapter 2 –.ppt

Chapter 2 – Signals, Systems & Fourier theory

• Concepts of signals, orthogonal function, Fourier theory, and correlation for spectral analysis.

• Linear systems & the impact of noise in the transmission of data

• By the end of this chapter: Goal: Understand signals and data representation in both time & frequency domain

Signals • Signal: Any time varying quantity that can be used to carry

information.

ASCII Encoding of A

• A 65 1000001 Signal TX

Classification of Signals (1)

• Based on 2 factors

• How it is represented in time

• How its amplitude is allowed to vary

This axis is continuous or discrete

Classification of Signals (2)

• The 4 Basic types of signals are:

• Continuous time, cont. amplitude• Continuous time, discrete amplitude• Discrete time, continuous amplitude• Discrete time, discrete amplitude

Classification of Signals (3)• The 4 Basic types of signals are:

Continuous-time Vrs Discrete-time

This axis is continuous or discrete

Continuous-time Signal(Sinusoid)

)()( tACostx

2T

Discrete-time Signal (1)(Derived from Cont. signal)

• Explanatory Notes on Sampling Theory:

• ‘Exact reconstruction of a continuous-time baseband signal from its samples is possible if the signal is bandlimited and the sampling frequency is greater than twice the signal bandwidth’.

Note: AnalogueDigital Conversion

• Analogue to Digital Conversion requires 3 essential steps:

• 1. Sampling: • 2. Quantization• 3. Encoding

Discrete-time Signal (2)

• Defined only @ discrete times• E.g., Exam results per semester• S (n) where n = …, -1, 0, 1, …, and are

functions defined on integers.

Analog vrs Digital

Periodic vrs Aperiodic

)()( tTftf )()( NnxnX

Examples of PeriodicSignals

Causal vrs Anti-Causal

Even vrs Odd Signals

Even-Odd Decomposition (1)

• Given a function X (t) of a signal;

)()(2

1)( txtxtxEv

)()(2

1)( txtxtxOdd



Class Work (5 mins)

• Given the functionx(t) = 2t + 1

Use the odd-even decomposition concept to show that

X(t) = Sum of Ev(x(t)) and Odd(x(t))

Deterministic vrs Stochastic

Random Signals (Noise)

n(t)

n(t)

0 t

2

2

)(1lim

)(

T

T

dttnTT

tn

2

2

22 1limT

T

dttnTT

tn

The square root of n^2(t) is the rms value of n(t).

Random Signals (Noise)

• Probability Density Function p(x)

• Probability that random variable lies b/n x1 and x2:

n(t)

n(t)

0 t

xxxxPxp oo )(

2

1

)(21

x

x

dxxpxxxP

Systems • Signals are always associated with one or more

systems

Systems Analysis (1)

Characterization of systems is by how many inputs and outputs they have:

• SISO (Single Input, Single Output) • SIMO (Single Input, Multiple Outputs) • MISO (Multiple Inputs, Single Output) • MIMO (Multiple Inputs, Multiple Outputs)


Systems could also be categorized on basis of type of signals:

• Analog System (Analog Input/ Analog Output) • Digital System (Digital Input/ Digital Output)

• Systems with Analog Input/ Digital Output or Vice versa


Another approach is on whether the systemhas memory or otherwise!

• Memoryless systems do not depend on any past input. (In digital electronics – Combinational Logic)

• Systems with memory do depend on past input. (In digital electronics – Sequential Logic)

• Causal systems do not depend on any future input.


Finally, systems are categorized by other properties such as:

• A system is linear if it has the superposition and scaling properties.

• A system that is not linear is non-linear. • If the output of a system does not depend explicitly on time,

the system is said to be time-invariant; otherwise it is time-variant

• A system that will always produce the same output for a given input is said to be deterministic.

• A system that will produce different outputs for a given input is said to be stochastic.

Linear Systems

• Linear systems must satisfy both homogeinity and additivity requirements:

• These 2 rules referred to as the principle of superposition

• Additivity: • Homogeneity:

Linear Systems

• Linear systems must satisfy both homogeinity and additivity requirements:

• These 2 rules referred to as the principle of superposition

• Additivity: • Homogeneity:

Fourier Transform

Signal – to – Noise Ratio

PowerNoise

PowerSignal

P

PSNR

noise

signal

noise

signal

noise

signal

A

A

P

PdBSNR 1010 log20log10)(

SNR & Capacity Cal. (Classwork)

• Given that the SNR of a channel is 3dB. How many bits can be transmitted in 1-hour for a given bandwidth of 30kHZ.

Correlation

• Correlation is a measure of how related two entities are.

• A high correlation means that there is a lot of resemblance between the two compared entities.

Auto-Correlation

• The auto-correlation for a periodic signal of period T is defined as follows:

• It defines how much a function correlates with a time shifted version of itself, with respect to that time shift.

2

2

)()(1

)(

T

Tiii dttWtW

TR

Cross-Correlation

• The cross-correlation for periodic signals of period T is defined as:

• It measures how much two different signals, Wi and Wj, one shifted in time with respect to the other, correlate as a function of that time shift.

2

2

)()(1

)(

T

Tjiij dttWtW

TC

Orthogonality

• Two periodic signals of period T are orthogonal when their cross- product is null for a zero time shift.

• Two orthogonal signals can be transmitted at the same time and will not interfere with each other. This principle is largely applied in CDMA.

2

2

0)()(

T

Tji dttWtW

Orthogonality

• The vectors (1, 3, 2), (3, −1, 0), (1/3, 1, −5/3) are orthogonal to each other, since (1)(3) + (3)(−1) + (2)(0) = 0, (3)(1/3) + (−1)(1) + (0)(−5/3) = 0, (1)(1/3) + (3)(1) − (2)(5/3) = 0.

• These vectors are orthogonal, for example (1, 0, 0, 1, 0, 0, 1, 0), (0, 1, 0, 0, 1, 0, 0, 1), (0, 0, 1, 0, 0, 1, 0, 0)

Power Spectral Density (PSD)

• PSD, describes how the power (or variance) of a time series is distributed with frequency.

• Mathematically, it is defined as the Fourier transform of the auto-correlation sequence of the time series.

• The term white noise refers to a noise whose power is distributed uniformly over all frequencies. White noise has a flat PSD.

Thermal Noise

• …caused by the random motion of molecules at any temperature above absolute zero Kelvin.

• Since the 3rd law of thermodynamics prevents one from extracting all heat from a physical system, one cannot reach absolute zero and so cannot entirely avoid thermal noise.

Time & Frequency Domain Rep

• Signals can be manipulated (i.e., amplified, filtered, etc.) in the time domain.

• However, it is often convenient and frequently necessary, when signal analysis and processing is required, to represent the signal in the frequency domain.



Time & Frequency Domain RepAssignment - 1

• Plot the time and frequency domain representation of the following signals:

Mathematical Representation Of Signals in Freq. Domain

• The theory of complex numbers is essential in understanding frequency domain representation. Revision

• In the ff sections, the concepts of Fourier analysis will provide us with a powerful tool for the general transformation of a signal from the time to frequency domain & the inverse transform!

Euler’s Identity

Complex Nos. Examples

Complex Nos. (Solve)

Complex Nos. (Solution)

Fourier Transform (1)

Joseph Fourier

• Joseph Fourier submitted a paper in 1807 to the Academy of Sciences of Paris. The paper was a mathematical description of problems involving heat conduction, and was at first rejected for lack of mathematical rigour. However, it contained ideas which have developed into an important area of mathematics named in his honour, Fourier analysis.

Fourier Transform (2)

Classification of signals ..• Energy and power signals

– A signal is an energy signal if, and only if, it has nonzero but finite energy for all time:

– A signal is a power signal if, and only if, it has finite but nonzero power for all time:

– General rule: Periodic and random signals are power signals. Signals that are both deterministic and non-periodic are energy signals.

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Chapter 2 –.ppt