My IntroductionJAHANGIR AHSANAsst. ProfessorBS in computer Engineering,SSUETMS in Network & Communication System ,Hamdard Univeristy Experience; 09 years, which include 01 years of field experience

My Introduction Research Interest : Network and Routers Devices. Office contactjahsan@ssuet.edu.pkJahangir_n@hotmail.comRoom no.ISPPhone; 4988000 / ext 373

Communications SystemsCE-3085TH. SEMESTERCOMPUTER ENGINEERING DEPARTMENT

COURSE OUTLINEIntroduction to Signals & Comm.Week 1Signal & Its ClassificationImportant SignalsOperations On SignalPower SignalsWeek 2Communication System ComponentsTypes of Communication SystemsApplication of Communication SystemSNR, Noise, Channel Bandwidth, Data rateBasic Filter TypesNyquist Theorem, Shannons Theorem

FOURIER ANALYSISWeek 3Fourier series for Periodic SignalsFourier TransformParsevals TheoremApplication of Fourier TransformPower Spectrum Density FunctionWeek 4Properties of Fourier TransformChannel & their types with distortions

FOURIER ANALYSISWeek 4 Convolution OperationImpulse Response & T.F of Channels/Filters

AMPLITUDE MODULATION AMWeek 5Double Side Band Suppressed Carrier(DSB-SC)Modulators & Demodulators of DSB-SC, DSB Single Band ModulationHilbert TransformWeek 6Modulators & Demodulators of SSBTelephone Channels using SSBVestigial Sideband VSB & ApplicationCarrier Acquisition & Phase locked loop PLL

INTRODUCTION TO DIGITAL COMMUNICATION (Week 7 & 8)

Digital Signal, Digital Communication ComponentSampling & QuantizationAWGN probability Density Function

BASEBAND DIGITAL COMMUNICATION(Note; Week 9 is Midterm)Week 10Detection of Baseband SignalsDetection 7 likelihood testWeek 11Vector representation of SignalsVector Analysis of Signals & noiseWeek 12Error probability & Error performance curvesMatch filters & Correlation

BAND PASS SIGNALS Week 13ASK, PSK & FSKDetection of Binary Band Pass Signals Detection of Multiple Band Pass Signals Week 14Different PSK ImplementationNon Coherent & Coherent Detection of SignalsOrthogonality of SignalsWeek 15Error performance for different modulation SchemesM-ary Signaling & PerformanceWeek 16 (Revision)

Text and Ref. BooksText BooksModern Digital and Analog Communication systems, 3rd. Edition by B.P.LathiDigital communication fundamental and Application, 2nd. Edition by Bernard Skalar Reference BooksSignals & Systems, 2nd. Edition by Alan V. Oppenheim and Alan S. Willsky with S. Hamid Nawab

Reference books continueDigital Signal Processing, Principles, Algorithms and Application by John G. Proakis and Dimitris G. Manolakis

Marks distributionTotal marks 100Theory (20)Attendance (2)Quizzes and Assignments (3)Midterm (15)Lab (20)Attendance (4)Lab manual (8)Final test (8)Semester Exam (60)

Signals and Systems(Week 1)In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying quantity.In the physical world, any quantity measurable through time can be taken as a signal.

Signals and Systems .Whereas the systems respond to particular signals by producing other signals or some desired behavior.For e.g. In Electrical circuit, voltages and currents as a function of time are signals, and a circuit is itself a system.Another example a robot arm is a system, whose movements are the response to control input signal.

Signals and SystemsThe concept of signals and systems arise in a wide variety of fields.Areas communications, aeronautics and astronautics, circuit design, acoustics, seismology, biomedical engineering, energy generation and distribution systems, chemical process control, and speech processing. Graphical rep. of a signal is shown in fig. 1.

Fig.1.Graphical rep. of Signal

Mathematical rep. of a signalS(t) = 8t ------------------------------------ (1)The above equation will given only linear relationship between signal S(t)and time t.S(t) = A Sine(2ft + ) ------------------(2)However, equation 2 is a sine wave or sinusoidal S(t) with parameters such as amplitude A, frequency f, time t and phase angle .Hence equation 2 is useful in analysis for e.g. speech signal, because it gives complete information about the signal compare to equation 1.

Math. Rep of a Signal Sine wave or SinusoidFig. 2, Phase diff of 900, Cosine wave leads by 900 A, theamplitude, is the peak deviation of the function from its center position. or 2f , theangular frequency, specifies how many oscillations occur in a unit time interval, inradiansper second, thephase, specifies where in its cycle the oscillation begins att= 0.The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase.

Math. Rep of a SignalIt is the only periodic waveform that has this property. This property leads to its importance inFourier analysisand makes it acoustically unique.Cosine wave said to be sinusoidal, because cosx = sin(x + /2), which is also a sine wave with a phase difference of /2.

Fig. 3.Occurrence of a Sinusoid

Classification of signalsPeriodic SignalA periodic signal repeat itself on a fixed interval of length 2 as shown in Fig. 4 . Sinusoidal signals are examples of it.

S(t) = S(t + T)Non-periodic SignalWhile non-periodic signal does not repeat itself as shown in Fig. 4.

Fig. 4 Classification of SignalsPeriodic Sig., with two PeriodsNon-Periodic Signal

(a) A periodic signal with period T0

(b) An aperiodic signal

Figure 3

Classification of signalsContinuous timeContinuous time signal or analog signal are defined for every value of time and they take on values in the continuous interval ( x, y)Discrete timeDiscrete time signal are defined only at certain specific values of time. These time instant needs not be equidistant. For e.g. n=0, +1, +2,..

Classification of SignalAnalog vs. DiscreteSame as Continuous vs. Discrete

Classification of SignalsDeterministicSignals whose values at any instant t is known from their analytical or graphical description are called deterministic signals, as shown in Figure 5.This type of signal convey no information.Example is electronic circuits based on Ohms law.RandomRandom means un-certainty. If the signal or message has un-certainty, it means, there is information as shown in Figure 6.Noise signals that perturb information are example of random signals.

Classification of signalsRandom Signal Fig.6Deterministic Signal Fig.5

Classification of SignalsCausal signals are signals that are zero for all negative time

Anti-causal are signals that are zero for all positive time.

Classification of signalsA signal fe(t) is said to be even, if it is identical to its time reversed counter part fe(t)=fe(-t)A signal fo(t)is said to be odd, if it is 0 at t=0; fo(t)=-(fo(-t))

(a) An even signal

(b) An odd signal

Classification of signalsContinuous time complex exponential (Fig.7)If is +, then as t increase s(t) is growing exponential.Examples; Chain reaction in atomic explosion, complex chemical reaction etcIf -, then s(t) is a decaying exponential.Examples; radioactive decay, Responses of RC circuit etc

Classification of signalsFig. 7 Exponential signals

(a) Ifis negative, we have the case of a decaying exponential window(b)Ifis positive, we have the case of a growing exponential window.(c)Ifis zero, we have the case of a constant window.

Real Exponential Signal:Real Exponential SignalWhere both"A"and""are real. Depending on the value of "" the signals will be different.

*

*The complex exponential signal is given by:

Where"s"is a complex variable and it is defined as

Complex exponential SignalUsing Eulers identitySubstituting eqn.(2) in eqn.(1) we haveEq. 1Eq. 2

(a) Ifis negative, we have the case of a decaying exponential window(b)Ifis positive, we have the case of a growing exponential window.(c)Ifis zero, we have the case of a constant window.

Signal Operations(Time Shifting of a Signal)Note; Subtracting from the time variable t will cause delay (move the signal to the right), while adding advance (to the left).

Signal OperationsTime Scaling

Time scaling compresses or dilates a signal by multiplying the time variable by some quantity. If that quantity is greater than one, the signal becomes narrower and the operation is called compression, while if the quantity is less than one, the signal becomes wider and is called dilation.

Signal OperationsTime ScalingIn general, if g(t) is compressed in time by a factor a (a> 1), the resulting signal is given by

Using the similar argument, we can show that g(t) expanded in time by a factor a (a

Time Scaling

Signal operationsTime ReversalA natural question arises about time scaling is: What happens when the time variable is multiplied by a negative number? The answer to this is time reversal. This operation is the reversal of the time axis, or flipping the signal over the y-axis.

Unit Impulse functionThe unit impulse function (t) is most of the important function in the study of signal and systems. This function was first defined by P.A.M. Dirac as(t) = 0 ; t0

Unit Impulse functionWe can visualize an impulse function as tallow rectangular pulse of unit area. The width of the rectangular pulse is some very small value. Its height is very large value. The unit impulse therefore can be regarded as a rectangular pulse with width that has become infinitesimally small, a height that has become infinitely large and an overall area that has been maintained unity.

Unit Impulse functionThus (t) = 0 everywhere except at t=0.

0t(t)

Signal Energy and Power In many applications the signals we consider are directly related a physical quantities capturing power and energy in physical system.

Signal Energy and Power The total energy expended over the time interval is

Similarly, for average power

Signal Energy The previous equations are basic, but they may be applied to continuous signal for evaluating Energy and Power, with the assumption that R=1.We may consider the area under the signal S(t) as a possible measure of its size, because it takes amplitude and duration both.This could be a improper measure, due to its large size and its positive and negative value, which cancel each other

Signal Energy signal is a function of varying amplitude through time, so a good measurement of the strength of a signal would be the area under the curve.However, this area may have a negative part. This negative part does not have less strength than a positive signal of the same sizeThe negative part cancels the positive

*This suggests either squaring the signal or taking its absolute value, then finding the area under that curveFigure 1Figure 2

Signal Energy This indicates a signal of small size.This difficulty can be corrected by defining signal size as S2(t), which is always positive.This measure is called Signal Energy. For a real valued and complex signal:-

Signal EnergyThe Signal Energy must be finite for it to be meaningful measure of the signal size. A necessary condition for the energy to be finite is that the signal amplitude 0 as |t|, otherwise the integral will not converge.

Signal PowerIf the amplitude of a signal does not 0 as |t|, the signal energy is infinite. In such a case a better measure of a signal size would be a, average power Ps, defined for a real and complex signal:-

Communications Systems(Week 2)Communication means to share information; To transmit or receive information or data. Communication may be established between people, Computer to Computer, near distance far distance (Telecommunication)The communication system , consist of different components which helps in transmitting or receiving data, between two entities or devices.

Communications SystemsAs shown in Fig. 8; Communication system components consists of Source, Input transducer , Transmitter, Channel, Receiver, and Output transducer. Source originates the message, for e.g. voice, picture or data. If the data is non-Electrical such as voice, then it must be converted to Electrical signal by Transducer known as baseband signal or message signal.

Fig. 8 Communication System

Input TransducerTransmitterChannelNoiseReceiverOutput transducerInput Signal Input Message TransmittedSignalReceivedSignalOutput SignalOutputMessage

Communications SystemsThe transmitter modifies the baseband signal for efficient transmission.A transmitter consists of one or more sub-systems, a sampler, a quantizer, a coder and a modulator.A channel is a medium, such as coaxial cable, a waveguide, an optical fiber, or a radio link-through which the transmitter output is sent.

Communications SystemsThe receiver reprocesses the signal from the channel by undoing the signal modification made at the transmitter and the channelFinally, the receiver output is fed to the output transducer, which convert back the electrical signal to its original form i.e. the message signal The Signal is distorted by Channel and Noise, which are random and unpredictable

Communications SystemsThat comes from external and internal sourcesExternal sources comes from nearby channel, lightning, tube light electrical equipment etcInternal noise results from thermal motions of electrons in conductors.The signal to noise ratio is defined as the ratio of the signal power to the noise power. The channel distort the signal and the noise accumulates along the path.

Communications SystemsThe signal strength decreases while the noise level increases with distance from the transmitter. Thus SNR is continuously decreasing along the channel and amplification of the noisy signal make no use.For good results SNR, supposed to be high. In other words signal value, should be high compare to noise.

Digital Communication SystemBlock diagram of CDMA system is shown in next slide. Since the human speech is the analog signal, so it has to be first converted into digital form. This function is performed by the source encoding module. After the source information is coded into a digital form, redundancy needs to be added to this digital message or data. This done for error control and power reduction.

Digital Communications SystemsBlock diagram

Source Encode

Modulate

MultipleAccess

ChannelEncode

Speech

Transmitter

Source Decode

De-Modulate

MultipleAccess

ChannelDecode

ReconstructedSpeech

Receiver

Digital Communication systemThereafter, signal is further transformed to allow access to multiple users. Multiple access by different users means to the sharing of a common resource i.e. RF (Radio Frequency) spectrum.The purpose of the modulator is to shift the message or data to the carrier frequency (high frequency), because message or data does not have enough strength to go far distance. At the receiver end the purpose of the demodulator is to recover original signal. In other words at the receiver end the reverse operation is performed.

Application of Comm. SystemTelephone ExchangeWireless Communication

Digital vs. AnalogDigitalMore immune to channel noise and distortionRegenerative repeaters for noise free signal.Digital hardware implementation is flexible in reconfiguring the hardware simply by changing the program.Accuracy

AnalogLess immune to noise and distortionNot possible in analog communicationFixed, and need to redesign for new hardware

Difficult to control the accuracy

Digital vs. AnalogDigitalCoding to yield low error rateEfficient in exchange of SNR for BandwidthAnalogLess exchange of SNR for Bandwidth

SNR, Bandwidth, Data RateThe fundamental parameters that control the rate and quality of information transmission are the channel bandwidth and the signal power S.The bandwidth (BW) of a channel is the range of frequencies that it can transmit with reasonable fidelity . ORDifference between the highest and the lowest frequencies in the specific range of frequencies.Example; Voice frequency range of 300hz. To 3300hz. Thus the voice bandwidth (BW) or Pass band is 3000hz wide

SNR, Bandwidth, Data Rate Role of BW, If we want to increase the speed of information transmission by time compression of the signal lets say by a factor of 2.The signal can be transmitted in half time.Frequencies and channel BW must also be doubled.Thus the rate of information transmission is directly proportional to channel BW. The signal power S plays a dual role in information transmission. Increase S reduced the effect of channel noise and we received accurate data.

SNR, Bandwidth, Data Rate Signal to Noise Ratio (SNR) means, the higher (strength) the value of the signal, compare to Noise, the quality of the signal would be better over a longer distance.However, a certain minimum SNR is necessary for communication.The second role of signal power S is not as obvious, but it is important. The BW and S are exchangeable.

SNR, Bandwidth, Data Rate For example if we increase the BW (by adding redundant bits to the message for reliability) the signal power S would reduced and vice versa. Another example; telephone channel has limited BW, but requires lot of power. So, we have to trade in between BW & S. Since SNR is proportional to S, Therefore SNR and BW are exchangeable. In practice, increasing BW to reduce signal power S is followed and is rarely vice versa.

Shannons and Nyquist TheoremThe limitation imposed on communication by the channel bandwidth BW and the SNR is highlighted by Shannons equation:-

Where C is the channel capacity in bits per second. This is the max. number of bits that can be transmitted per second with a probability of error close to zero.

Shannons and Nyquist TheoremIf there is no noise in the channel, N=0 .The Nyquist Theorem, also known as the sampling theorem, is a principle that engineers follow in the digitization ofanalogsignals. For analog-to-digital conversion (ADC) to result in a faithful reproduction of the signal, slices, calledsamples, of the analogwaveformmust be taken frequently. The number of samples per second is called the sampling rate or sampling frequency.

Nyquist TheoremAny analog signal, consists of components at various frequencies. The simplest case is thesine wave, in which all the signal energy is concentrated at one frequency. In practice, analog signals usually have complex waveforms, with components at many frequencies. The highest frequency component in an analog signal determines thebandwidthof that signal. The higher the frequency, the greater the bandwidth.

Nyquist TheoremSuppose the highest frequency component, inhertz, for a given analog signal isfmax. According to the Nyquist Theorem, the sampling rate must be at least 2fmax, or twice the highest analog frequency component. If the sampling rate is less than 2fmax, some of the highest frequency components in the analog input signal will not be correctly represented in the digitized output. When such a digital signal is converted back to analog form by a digital-to-analog converter, false frequency components appear that were not in the original analog signal. This undesirable condition is a form of distortion calledaliasing.

Examples of Aliasing1. Consider two sinusoidal signals:-

Which are sampled at a rate The corresponding discrete signals are

Example of Aliasing Result is that after digitizing, the value of signal s2 (5 integer multiple i.e. cos /2)is identical with s1 (cos/2) . In other words frequency of signal s2 is alias with the frequency of signal s1.So, as per Nyquist criteria, sampling rate Fs >2fmax of the signal components. In this case Fs should be 100hz. i.e. twice of 50hz.

Examples of Aliasing2. Consider the analog signal:-

What is the Nyquist rate for this signal?Solution; F1,F2,F3 are 25,150,50hz. Respectively.(Note If the signal for e.g. is cos2(10)t, no need to divide it by 2. Because it is already in the form of 2f. In the other case, for e.g. cos 50t , in order to make it in the form of 2f, we have to divide frequency 50hz by 2.

Aliasing ExamplesThus fmax is 150hz, therefore Fs >2fmax =300hz

Basic Filter TypesA filter is a circuit that is designed to pass a specific band of frequencies, while block all signals outside this band.Application include (but certainly not limited to) noise rejection, signal separation, smoothing of digitally generated analog signals, audio signal shaping etcThere are four types of filter; low-pass, high-pass, band-pass, and band elimination also known as notch filter.

Basic filter types Alow-pass filteris anelectronic filterthat passes low frequencysignalsbutattenuates(reduces theamplitudeof) signals with frequencies higher than thecutoff frequency Fig.1.Ahigh-pass filter(HPF) is a device that passes highfrequenciesandattenuates(i.e., reduces the amplitude of) frequencies lower than itscutoff frequency Fig.1.

Basic filter typesAband-pass filteris a device that passesfrequencieswithin a certain range and rejects (attenuates) frequencies outside that range Fig.2.A Notch filter is a filter that passes all frequencies except those in a stop band centered on a center frequency Fig.2.Notch filters are used to reject unwanted signals for e.g. spikes in sensitive instruments.

Fig.1. Basic Filter typesLow Pass filterHigh pass filter

Fig.2.Basic Filter typesBand pass filterNotch filter

Week 3 and 4Fourier SeriesFourier Transform

Time & Frequency domain conceptTime-domain graph shows how a signal changes over timeIn the freq. domain, all the component s (freq. and amplitude) of a sine wave can be represented by a single vertical line. Similarly, freq. and phase, as shown in Fig. 4 Frequency domain is a term used to describe the domain for analysis of signals with respect to frequency, rather than time.Frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift.

Time & Frequency domain concept.Multiple sine wave called a composite signal, which is used in communication.ExampleS(t) =1 sine10(100000t+ )Where 1 is the amplitude in volts, 10 is the no. of cycles, (angular frequency) = 2f , so f = 50khz, t is the time and is the phase, in this case it is 0, because sine wave starts at 0 origin.Graphical rep. of this equation is shown in fig.4.

Fig. 4 Time & Frequency domain example10 periods of 1 Volt, 50KHz sine waveAmplitude in the Single Sided DFT in the 50KHz bin is 1.0 V

Time & Frequency domain conceptA given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called a transform. An example is the Fourier transform, which decomposes a function into the sum of a (potentially infinite) number of sine wave frequency components. The 'spectrum' of frequency components is the frequency domain representation of the signal. The inverse Fourier transform converts the frequency domain function back to a time function.

Fourier Analysis Fourier analysis provides us a way to view time domain signal in frequency domain. Fourier series and Fourier Transform are part of it. The former deal with periodic signal, which can be represented as a sum of sinusoids as shown in fig Fig. 5. Actually, this figure shows the verification of conversion of square wave function into Fourier series. While the latter is related with non-periodic signals. This is shown in Fig.6.

Fourier Analysis Signals like pulse and transient are of practical importance in communications. These signals cannot be analyzed by Fourier series due to mathematical constraints.The Fourier Transform decomposes a waveform - basically any real world waveform, into sinusoids. That is, the Fourier Transform gives us another way to represent a waveform. The Fourier series is named in honor of Joseph Fourier (17681830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d Alembert, and Daniel Bernoulli

Fig. 5 Fourier series approximations for a square wave.

Fourier SeriesInmathematics, aFourier seriesdecomposesperiodic functionsor periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namelysines and cosines(orcomplex exponentials).Therefore, a signal g(t) can be expressed by a trigonometric Fourier series over any interval of duration T0 second as

Trigonometric Fourier Series

Trigonometric Fourier SeriesWhere , for the coefficients a0,an and bn we have

Compact Trigonometric Fourier SeriesThe trigonometric Fourier series contains sine and cosine of the same frequency. We can combine the two terms in a single term of the same frequency using the trigonometric identity:-

Compact Trigonometric Fourier Series

Fourier TransformTheFourier transformis a mathematical operation with many applications inphysicsandengineeringthat expresses a mathematicalfunctionof time as a function offrequency, known as itsfrequency spectrum;The Fourier Transform is used for non-periodic or Aperiodic signals as shown in Fig.6.The function oftimeis often called thetime domainrepresentation, and the frequency spectrum thefrequency domainrepresentation. The inverse Fourier transform expresses a frequency domain function in the time domain. Each value of the function is usually expressed as a complex number (calledcomplex amplitude) that can be interpreted as a magnitude and a phase component.

Fourier TransformFig. 6;Aperiodic signal

Fourier TransformThe representation of non-periodic signals by eternal exponential (everlasting or endless) can be accomplished by a simple limiting process. In other words non-periodic signals can be expressed as a continuous sum (integral) of a signal, which have no duration.This is shown in Fig.7a & 7b.(Note; Short table 3.1 of Fourier Transforms is given on page 85 of 3rd. Edition)

Figure 7a&b; Aperiodic signal and its representation as a continuous sum.

Fourier TransformAs shown in figure 7b a new periodic signal gp (t)consisting of signal g(t) repeating itself every T0 sec. This period is made long enough so that there is no overlap between the repeating pulses.This new signal gp (t) is a periodic signal and can be represented by Fourier series. (Note derivation is given in the B.P. Lathi book)

Fourier TransformThe Fourier transform and its inverse of Aperiodic signal in terms of eternal exponential function is given below respectively:-

Fourier TransformWhere G() is the frequency domain representation of the continuous signal g(t), with frequencies lying in the interval (-

Properties of Fourier TransformReciprocity of signal Duration and its Bandwidth. This suggests that the bandwidth of a signal is inversely proportional to the signal duration or width (in seconds)Time shifting propertyg(t) G()g(t-t0)

Properties of Fourier Transform

Symmetry propertyg(t) G()Then G(t) 2g(-)Scaling propertyIf g(t) G()Then g(at)

Properties of Fourier TransformFrequency-Shifting Property g(t) G()

Transforms of some useful functionsUnit gate functionIt is define as a unit gate function rect(x) as a gate pulse of unit height and unit width, centered at the origin rect(x) = 0 if x>1/2 rect(x) = 1/2 if x=1/2 rect(x) = 1 if x

Transforms of some useful functions.. Interpolation function Sinc(x)

Explanation of Sinc Pulse:-Sinc; Sine cardinal; cardinal means; fundamental- Sinc(x)=0,when Sine (x)=0 , for x= ,2,3 except at x=0 ;i.e. Sinc(0) =1 (proof given below)Sinc(0) = Sine (0)/0 = 0/0 (indeterminate) -Using L Hopitals rule; Sinc (0)=1 as follows:-Sinc(x)= Lim (x=0) ;Sine (x)/x eq.(1)Differentiate eq.(1) w.r.t to x ,Lim(x=0) Cos (x)/1Cos (0)/1 = 1/1 = 1So,Sinc(0)=1

Application of Sinc Pulse: - Pulse shaping is used to improve spectral efficiency. Suppose we want to increase transmission speed without reducing accuracy or increasing band width e.g. faster telephone modem for internet. The band width of telephone line is fixed. We need a pulse that required less bandwidth and less ISI (inter symbol interference).- We may use Sinc pulse to improve spectral efficiency

Parsevals TheoremDef.1: Energy of a signal in time domain is equal to the square of the spectrum in the frequency domain.Def.2: If the signal is assumed to be a voltage and it is applied across 1 ohm resistor, then the energy dissipated in the resistor is given by the left hand side of the equation below. Parsevals theorem states that this is also equal to integral of the square of the spectrum on the right.

Power Spectral DensityPower of the periodic signal is distributed among the various frequency components.Power spectral density function (PSD) shows the strength of the variations(energy) as a function of frequency. In other words, it shows at which frequencies variations are strong and at which frequencies variations are weak. The unit of PSD is energy per frequency(width) and you can obtain energy within a specific frequency range by integrating PSD within that frequency range.

Power Spectral DensityComputation of PSD is done directly by the method called FFT or computing autocorrelation function and then transforming it.

Power Spectral DensityWe define the power spectral density (PSD) () as

Where GT(), is signal spectrum and T is the time.

Channel & their types with distortionsThe channel is a medium such as wire, co-axial cable a wave guide, an optical fiber, or a radio link through which the transmitter output is sent.Distortion; spreading or dispersion of the pulse will occur if either the amplitude response or the phase response (linear distortion) or both are not ideal. For example in TDM, pulse spreading causes interference.

Channel & their types with distortionsPlease note that linear distortion is valid only for small signals. For large amplitudes, non-linearity cannot be ignored for example memory less channel.

Impulse Response:

Convolution

The convolution of two signals or functions g(t) and w(t), denoted as follows:-

g(t) * w(t) =

Application is in digital filtering

Convolution (detail):It is an operation on two functions or signals. Lets say g(t) ,w(t) producing a third function that is typically viewed as a modified ver. Of one of the original function or signal, giving the area overlap below the two signals as a function of the amount that one of the original function is translated. Convolution is similar to cross correlation. It has application that include probability, statistics, image signal processing etc.

Comparison among Convolution, Cross-correlation and Autocorrelation

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