signals and systems 8

Signals and SystemsLinear System TheoryEE 112 - Lecture Eight Signal classification

Khosrow Ghadiri - EE Dept. SJSU

SignalA signal is a pattern of variation that carry information.Signals are represented mathematically as a function of one or more independent variable A picture is brightness as a function of two spatial variables, x and y.In this course signals involving a single independent variable, generally refer to as a time, t are considered. Although it may not represent time in specific applicationA signal is a real-valued or scalar-valued function of an independent variable t.


Example of signalsElectrical signals like voltages, current and EM field in circuitAcoustic signals like audio or speech signals (analog or digital)Video signals like intensity variation in an imageBiological signal like sequence of bases in gene Noise which will be treated as unwanted signal


Signal classificationContinuous-time and Discrete-time Energy and PowerReal and ComplexPeriodic and Non-periodic Analog and DigitalEven and Odd Deterministic and Random


A continuous-time signalContinuous-time signal x(t), the independent variable, t is Continuous-time. The signal itself needs not to be continuous.


A continuous-time signal x=0:0.05:5; y=sin(x.^2); plot(x,y);


A piecewise continuous-time signalA piecewise continuous-time signal


A discrete-time signalA discrete signal is defined only at discrete instances. Thus, the independent variable has discrete values only.


Mathlab file x = 0:0.1:4; y = sin(x.^2).*exp(-x); stem(x,y)


SamplingA discrete signal can be derived from a continuous-time signal by sampling it at a uniform rate. If denotes the sampling period and denotes an integer that may assume positive and negative values,Sampling a continuous-time signal x(t) at time yields a sample of value For convenience, a discrete-time signal is represented by a sequence of numbers:We write

Such a sequence of numbers is referred to as a time series.


A piecewise discrete-time signalA piecewise discrete-time signal


Energy and Power SignalsX(t) is a continuous power signal if:

X[n] is a discrete power signal if:

X(t) is a continuous energy signal if:

X[n] is a discrete energy signal if:


Power and Energy in a Physical System The instantaneous power

The total energy

The average power


Power and EnergyBy definition, the total energy over the time interval in a continuous-time signal is:

denote the magnitude of the (possibly complex) numberThe time average power

By definition, the total energy over the time interval in a discrete-time signal is:

The time average power


Power and EnergyExample 1:The signal is given below is energy or power signal. Explain.

This signal is energy signal


Power and EnergyExample 2:The signal is given below is energy or power signal. Explain.

This signal is energy signal


Real and ComplexA value of a complex signal is a complex number

The complex conjugate, of the signal is;

Magnitude or absolute value

Phase or angle


Periodic and Non-periodicA signal or is a periodic signal if

Here, and are fundamental period, which is the smallest positive values when Example:


Analog and DigitalDigital signal is discrete-time signal whose values belong to a defined set of real numbers

Binary signal is digital signal whose values are 1 or 0

Analog signal is a non-digital signal


Even and Odd

Even Signals The continuous-time signal /discrete-time signal is an even signal if it satisfies the condition

Even signals are symmetric about the vertical axis Odd Signals The signal is said to be an odd signal if it satisfies the condition

Odd signals are anti-symmetric (asymmetric) about the time origin


Even and Odd signals:Facts

Product of 2 even or 2 odd signals is an even signalProduct of an even and an odd signal is an odd signalAny signal (continuous and discrete) can be expressed as sum of an even and an odd signal:


Complex-Valued Signal SymmetryFor a complex-valued signal

is said to be conjugate symmetric if it satisfies the condition

where

is the real part and is the imaginary part;

is the square root of -1


Deterministic and Random signalA signal is deterministic whose future values can be predicted accurately.Example: A signal is random whose future values can NOT be predicted with complete accuracyRandom signals whose future values can be statistically determined based on the past values are correlated signals.Random signals whose future values can NOT be statistically determined from past values are uncorrelated signals and are more random than correlated signals.


Deterministic and Random signal(contd)Two ways to describe the randomness of the signal are:Entropy: This is the natural meaning and mostly used in system performance measurement.Correlation: This is useful in signal processing by directly using correlation functions.


Taylor series and Euler relationGeneral Taylor series

Expanding sin and cos

Expanding exponential

Euler relation


ExampleThe signal and shown below constitute the real and imaginary parts of a complex-valued signal .What form of symmetry does have?

Answer is conjugate symmetric.


signals and systems 8

Documents