Signal and Information Processing

Presented by Huanhuan ChenUniversity of Science and Technology of ChinaSignal and Information Processing

1

Course informationCourse Homepage: http://staff.ustc.edu.cn/~hchen/signal

Email address: hchen@ustc.edu.cn

2An Introduction to Signal and Information Processing3About SignalThe term signal is generally applied to something that conveys information. The independent variable in the mathematical representation of a signal may be either continuous or discrete.

Voltage across a resisterVelocity of a vehicle Light intensity of an imageTemperature, pressure inside a system

41-d signalsSeismic vibrationsEEG and EKGSpeechSonarAudio Music

ph - o - n - e - t - i - c - ia - n

52-d signals.PhotographsMedical imagesRadarIED detectionSatellite dataFaxFingerprints

63-d signals.Video Sequences Motion SensingVolumetric data sets Computed Tomography, Synthetic Aperture Radar Reconstruction)

7An birds view of signal processingProcessing systemInformationSignalRecognition- radar, sonar, seismic, StorageTransmissionProcessing systemDisplayInformationCommunicationsStorage Media

8Continuous-Time signalContinuous-time signals are defined along a continuum of times and thus are represented by a continuous independent variable. Continuous-time signals are often referred to as analog signals.

9Discrete-time signalDiscrete-time signals are defined at discrete times, and thus, the independent variable has discrete values.Digital signals are those for both time and amplitude are discrete.Discrete-time signals are represented mathematically as sequences of numbers

10Discrete-time signal

11Some basic sequencesThe sequences shown play important roles in the analysis and representation of discrete-time signals and systems.

12Elementary SignalsSinusoidal & Exponential SignalsSinusoids and exponentials are important in signal and system analysis because they arise naturally in the solutions of the differential equations.Sinusoidal Signals can expressed in either of two ways : cyclic frequency form- A sin 2fot = A sin(2/To)t radian frequency form- A sin ot o = 2fo = 2/To To = Time Period of the Sinusoidal Wave13Sinusoidal & Exponential Signalsx(t) = A sin (2fot+ )= A sin (ot+ ) x(t) = Aeat Real Exponential = Aejt = A[cos (ot) +j sin (ot)] Complex Exponential = Phase of sinusoidal wave A = amplitude of a sinusoidal or exponential signal fo = fundamental cyclic frequency of sinusoidal signal o = radian frequencySinusoidal signal14 Unit Step Function

Precise GraphCommonly-Used Graph

15Signum Function

Precise GraphCommonly-Used Graph

The signum function, is closely related to the unit-step function.16Unit Ramp Function

The unit ramp function is the integral of the unit step function.It is called the unit ramp function because for positive t, its slope is one amplitude unit per time.17

Rectangular Pulse or Gate FunctionRectangular pulse,

18Unit Impulse Function

So unit impulse function is the derivative of the unit step function or unit step is the integral of the unit impulse function

Functions that approach unit step and unit impulse19Representation of Impulse FunctionThe area under an impulse is called its strength or weight. It is represented graphically by a vertical arrow. An impulse with a strength of one is called a unit impulse.

20Properties of the Impulse Function

The Sampling Property

The Scaling PropertyThe Replication Propertyg(t) (t) = g (t)21Unit Impulse TrainThe unit impulse train is a sum of infinitely uniformly-spaced impulses and is given by

22The Unit Rectangle FunctionThe unit rectangle or gate signal can be represented as combination of two shifted unit step signals as shown

23The Unit Triangle Function

A triangular pulse whose height and area are both one but its base width is not, is called unit triangle function. The unit triangle is related to the unit rectangle through an operation called convolution. 24Sinc Function

25Time Domain and Frequency DomainMany ways that information can be contained in a signal. Manmade signals. AMFMSingle-sideband Pulse-code modulation Pulse-width modulationOnly two ways that are common for information to be represented. Information represented in the time domain, Information represented in the frequency domain.

26The time domainDomain describes when something occursWhat the amplitude of the occurrence was Each sample in the signal indicates What is happening at that instant, and the Level of the event If something occurs at time t, the signal directly provides information on the time it occurred, the duration, and the development over time. Contains information that is interpreted without reference to any other part of the sample.

27The frequency domainFrequency domain is considered indirect.Information is contained in the overall relationship between many points in the signal.By measuring the frequency, phase, and amplitude, information can be obtained about the system producing the motion.28Converting analog to digital signalsSampling is the acquisition of the values of a continuous-time signal at discrete points in timex(t) is a continuous-time signal, x[n] is a discrete-time signal

29Signal samplingNyquist sampling theorem. The lower bound of the rate at which we should sample a signal, in order to be guaranteed there is enough information to reconstruct the original signal is 2 times the maximum frequency.Now in its digital form,we can process the signalin some way.

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30Sampling continuous signal

Sampling continuous signal

Discrete Time Unit Step Function or Unit Sequence Function

33Discrete Time Unit Ramp Function

34Discrete Time Unit Impulse Function or Unit Pulse Sequence

35Techniques for processing a signalA system is a function that produces an output signal in response to an input signal.

An input signal can be broken down into a set of components, called an impulses. Impulses are passed through a system resulting in output components, which are synthesized into an output signal.

Operations of SignalsSometime a given mathematical function may completely describe a signal .Different operations are required for different purposes of arbitrary signals.The operations on signals can be Time Shifting Time Scaling Time Inversion or Time Folding37Time ShiftingThe original signal x(t) is shifted by an amount t.

X(t)X(t-to) Signal Delayed Shift to the right

Time ShiftingX(t)X(t+to) Signal Advanced Shift to the left

Time ScalingFor the given function x(t), x(at) is the time scaled version of x(t)For a 1,period of function x(t) reduces and function speeds up. Graph of the function shrinks.For a 1, the period of the x(t) increases and the function slows down. Graph of the function expands.Time scalingExample: Given x(t) and we are to find y(t) = x(2t). The period of x(t) is 2 and the period of y(t) is 1,

41Example: fast forwarding / slow playing Time scalingGiven y(t), find w(t) = y(3t) and v(t) = y(t/3).

42Time ReversalTime reversal is also called time foldingIn Time reversal signal is reversed with respect to time i.e. y(t) = x(-t) is obtained for the given functionTime reversal Contd.

44Example: Playing a tape recorder backwardBeatle Revolution 9 is said be played backward!

Operations of Discrete Time Functions

45Operations of Discrete Functions

K an integer > 1

46Convolution of two sequence

(a)(c) The sequences x[k] and h[n k] as a function of k for different values of n. (Only nonzero samples are shown.) (d) Corresponding output sequence as a function of n.Properties of ConvolutionCommutative Property

Distributive Property

Associative Property

48FilteringDigital filtering: Digital filtering can be accomplished by number of different well-characterized numerical procedure (a) Fourier transformation (b) Least squares polynomial smoothing.

Fourier transformationFourier transformation: In this transformation, a signal which is acquired in the time domain, is converted to a frequency domain signal in which the independent variable is frequency rather than time. This transformation is accomplished mathematically on a computer by a very fast and efficient algorithm. The frequency domain signal is then multiplied by the frequency response of a digital low pass filter which remove frequency components. The inverse Fourier transform then recovers the filtered time domain spectrum.

Discrete Fourier transform Given the time domain, the process of calculating the frequency domain is called DFT.Given frequency domain the process of calculating the time domain is inverse DFT.O(n2)

Discrete Fourier transformDFT for continuous signals, not for digital signals.

DFTInverse DFTPlug in angular frequency f.DFT to get frequency.Inverse DFT to get time t.Discrete Fourier transformConvert continuous DFT to discrete DFT.Continuous versionDiscrete version

Let stand for (a primitive nth root of unity)We get

Smoothing of signal with DFTIntuitions of DFT

Boxes as Sum of Cosines

Introduction to System

What is System?Systems process input signals to produce output signalsA system is combination of elements that manipulates one or more signals to accomplish a function and produces some output.systemoutput signalinput signalExamples of Systems

A circuit involving a capacitor can be viewed as a system that transforms the source voltage (signal) to the voltage (signal) across the capacitorA communication system is generally composed of three sub-systems, the transmitter, the channel and the receiver. The channel typically attenuates and adds noise to the transmitted signal which must be processed by the receiverBiomedical system resulting in biomedical signal processingControl systems

System - ExampleConsider an RL series circuitUsing a first order equation:

LV(t)R60Mathematical Modeling of Continuous Systems Most continuous time systems represent how continuous signals are transformed via differential equations.E.g. RC circuit

System indicating car velocity

Mathematical Modeling of Discrete Time Systems Most discrete time systems represent how discrete signals are transformed via difference equationse.g. bank account, discrete car velocity system

Order of SystemOrder of the Continuous System is the highest power of the derivative associated with the output in the differential equationFor example the order of the system shown is 1.

Order of SystemOrder of the Discrete Time system is the highest number in the difference equation by which the output is delayedFor example the order of the system shown is 1.

Interconnected Systemsnotes

Parallel Serial (cascaded) FeedbackLV(t)RLC65Interconnected System ExampleConsider the following systems with 4 subsystemEach subsystem transforms it input signalThe result will be:y3(t)=y1(t)+y2(t)=T1[x(t)]+T2[x(t)]y4(t)=T3[y3(t)]= T3(T1[x(t)]+T2[x(t)])y(t)= y4(t)* y5(t)= T3(T1[x(t)]+T2[x(t)])* T4[x(t)]

66Feedback SystemUsed in automatic control e(t)=x(t)-y3(t)= x(t)-T3[y(t)]=y(t)= T2[m(t)]=T2(T1[e(t)]) y(t)=T2(T1[x(t)-y3(t)])= T2(T1( [x(t)] - T3[y(t)] ) ) ==T2(T1([x(t)] T3[y(t)]))

67Types of SystemsCausal & AnticausalLinear & Non LinearTime Variant & Time-invariantStable & Unstable Static & DynamicInvertible & Inverse Systems

Causal & Anticausal SystemsCausal system : A system is said to be causal if the present value of the output signal depends only on the present and/or past values of the input signal.Example: y[n]=x[n]+1/2x[n-1] Causal & Anticausal Systems Contd.Anticausal system : A system is said to be anticausal if the present value of the output signal depends only on the future values of the input signal.Example: y[n]=x[n+1]+1/2x[n-1]Linear & Non Linear SystemsA system is said to be linear if it satisfies the principle of superpositionFor checking the linearity of the given system, firstly we check the response due to linear combination of inputs Then we combine the two outputs linearly in the same manner as the inputs are combined and again total response is checkedIf response in step 2 and 3 are the same, the system is linear othewise it is non linear.Time Invariant and Time Variant SystemsA system is said to be time invariant if a time delay or time advance of the input signal leads to a identical time shift in the output signal.

Stable & Unstable Systems A system is said to be bounded-input bounded-output stable (BIBO stable) iff every bounded input results in a bounded output. i.e.

Stable & Unstable Systems Contd.Example - y[n]=1/3(x[n]+x[n-1]+x[n-2])

Stable & Unstable Systems Contd.Example: The system represented by y(t) = A x(t) is unstable ; A1

Reason: let us assume x(t) = u(t), then at every instant u(t) will keep on multiplying with A and hence it will not be bonded.Static & Dynamic SystemsA static system is memoryless systemIt has no storage devicesits output signal depends on present values of the input signalFor example

Static & Dynamic Systems Contd.A dynamic system possesses memoryIt has the storage devicesA system is said to possess memory if its output signal depends on past values and future values of the input signal

Example: Static or Dynamic?Example: Static or Dynamic?Answer:The system shown above is RC circuitR is memoryless C is memory device as it stores charge because of which voltage across it cant change immediatelyHence given system is dynamic or memory systemInvertible & Inverse SystemsIf a system is invertible it has an Inverse System

Example: y(t)=2x(t)System is invertible must have inverse, that is: For any x(t) we get a distinct output y(t)Thus, the system must have an Inversex(t)=1/2 y(t)=z(t)y(t)SystemInverse Systemx(t)x(t)y(t)=2x(t)System(multiplier)

Inverse System(divider)x(t)x(t)80LTI SystemsLTI Systems are completely characterized by its unit sample responseThe output of any LTI System is a convolution of the input signal with the unit-impulse response, i.e.

Useful Properties of (DT) LTI SystemsCausality:Stability:Bounded Input Bounded Output

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