## Analog Demodulation

Beam me up, Scotty. - attributed to James T. Kirk Several parts of a communications system modulate the signal and change the underlying frequency band in which the signal lies. These frequency changes must be reversible after processing, the receiver must be able to reconstruct (a close approximation to) the transmitted signal. FIGURE 5.1 Digital Communication System FIGURE 5.1 Digital Communication System The input message w(kT) in Figure 5.1 is a discrete-time sequence drawn from a finite...

## The component architecture layer

The next two chapters provide more depth and detail by outlining a complete telecommunication system. When the transmitted signal is passed through the air using electromagnetic waves, it must take the form of a continuous (analog) waveform. A good way to understand such analog signals is via the Fourier transform, and this is reviewed briefly in Chapter 2. The five basic elements of the receiver will be familiar to many readers, and they are presented in Chapter 3 in a form that will be...

## Modelling Corruption

From there to here, from here to there, funny things are everywhere. - Dr. Suess, One Fish, Two Fish, Red Fish, Blue Fish, 1960. If every signal that went from here to there arrived at its intended receiver unchanged, then the life of a communications engineer would be easy. Unfortunately, the path between here and there can be degraded in several ways, including mul-tipath interference, changing (fading) channel gains, interference from other users, broadband noise, and narrow-band...

## Y

FIGURE 12.9 One implementation of the adaptive element (12.13) uses three digital interpolations (resamplers). After the r k converge, the output x k is a sampled version of the input x(t), with the samples taken at times that maximize the power of the output. xs (i) interpsinc (x ,tnow+tau, 1) , interpolated value at tnow+tau , get value to the right x_deltam interpsinc(x,tnow+tau-delta,l) , get value to the left dx x_deltap-x_deltam , calculate numerical derivative tau tau+mu*dx*xs(i) , alg...

## E 1 1 HT E sitkmedt

N co Jt -oo fc -co If v(t) is a Nyquist pulse, the only nonzero term in the sum is i> (0), and Thus, the sum of the V(f nfo) is a constant if v(t) is a Nyquist pulse. Conversely, if the sum of the V(f nfo) is a constant, then only the DC term in (11.5) can be nonzero, and so v(t) is a Nyquist pulse. 11.14. Write a Hatlab routine that implements the raised cosine impulse response (11.4) with rolloff parameter 3. Hint If you have trouble with divide by zero errors, imitate the code in SRRC.m....

## Envelope Of A Bandpass Signal

You know that the Radio wave is sent across, transmitted from the transmitter, to the receiver through the ether. Remember that ether forms a part of everything in nature-that is why Radio waves travel everywhere, through houses, through the earth, through the air. -Fundamental Principles of Radio Certified Radio-tricians Course, National Radio Institute, Washington DC, 1914. The envelope of a signal is a curve that smoothly encloses the signal, as shown in Figure C.l. An envelope detector is a...

## Wf f J2 Wfnfs62

Thus the spectrum of the sampled signal ws (t) differs from the spectrum of the original w(t) in two ways Amplitude scaling each term in the spectrum Ws(f) is multiplied by the factor fs. Replicas for each n, Ws(f) contains a copy of W(f) shifted to nfs. Sampling creates an infinite sequence of replicas, each separated by fs Hz. Said another way, sampling-in-time is the same as periodic-in-frequency, where the period is defined by the sampling rate. Readers familiar with Fourier series will...

## Encoding A Compact Disc

The process of writing to and reading from a compact disc is involved. The essential idea in optical media is that a laser beam bounces off the surface of the disc. If there is a pit, then the light travels a bit further than if there is no pit. The distances are controlled so that the extra time required by the round trip corresponds to a phase shift of 180 degrees. Thus the light travelling back interferes destructively if there is a pit, while it reinforces constructively if there is no pit....

## Derivatives And Filters

Averages and low pass filters occur within the definitions of the performance functions associated with many adaptive elements. For instance, the AGC of Chapter 6, the phase tracking algorithms of Chapter 10, the timing recovery methods of Chapter 12, and the equalizers of Chapter 14 all involve LPFs, averages, or both. Finding the correct form for the adaptive updates requires taking the derivative of filtered and averaged signals. This section shows when it is possible to commute the two...

## Coding

Before Shannon it was commonly believed that the only way of achieving arbitrarily small probability of error on a communications channel was to reduce the transmission rate to zero. Today we are wiser. Information theory characterizes a channel by a single parameter the channel capacity. Shannon demonstrated that it is possible to transmit information at any rate below capacity with an arbitrarily small probability of error. from A. R. Calderbank, The Art of Signaling Fifty Years of Coding...

## Noisychanm generate 4level data and add noise

length of data sequence power of noise and signal generate 4-PAM input with power 1 generate noise with power p output of system adds noise to data quantize output to -3*1,-1,1,3*1 percent transmission errors Typical outputs of noisychan.m are shown in Figure 15.4. Each plot shows the input sequence (the four solid horizontal lines), the input plus the noise (the cloud of small dots), and the error between the input and quantized output (the dark stars). Thus the dark stars that are not at...

## Info

Thus if there are n letters, there are 7n (uncoded) bits, 7n( ) coded bits, and 7n( )( ) 4-PAM symbols. These mappings are familiar from Section 8.1, and are easy to use with the help of the Matlab functions bin2text .m and text2bin.m. Problem 15.25 provides several hints to help implement the M6 encoding, and the Matlab function nocode.m outlines the necessary transformations from the original text into a sequence of 4-PAM symbols s i . In order to decode the message at the...

## Performance evaiualiori

FIGURE 14.7 Dispersion Minimizing Adaptive Linear Equalizer When the source alphabet is 1, then 7 1. When the source is multilevel, it is still useful to minimize the dispersion, but the constant should change to While DMA typically may converge to the desired answer from a worse initialization than decision-directed LMS, it is not as robust as trained LMS. For a particular delay S, the (average) squared recovery error surface being descended (approximately) along the gradient by trained LMS is...

## The M6 Receiver Design Challenge

The analog front end of the receiver in Figure 16.2 takes the signal from an antenna, amplifies it, and crudely bandpass filters it to (partially) suppress frequencies outside the desired user's frequency band. An analog converter modulates the received signal (approximately) down to the nominal intermediate frequency at 2 MHz. The output of the analog downconverter is adjusted by an automatic gain controller to fit the range of the sampler. The output of the AGC is sampled at intervals of Ts...

## Other Impairments More What

Of course, a fading channel is not the only thing that can go wrong in a telecommunications system. (Think back to the what if questions in the first section of this chapter.) This section considers a range of synchronization and interference impairments that violate the assumptions of the idealized system. Though each impairment is studied separately (i.e., assuming that everything functions ideally except for the particular impairment of interest), a single program is written to simulate any...

## R2t 12 s2 i [1 cos47roi

Rewrite -s2(t) as the sum of its (positive) average value and the variation about this average r2(t) (lf2) s2avg + v(t) + s2avgcoS(A7Tf0t + 20) + t,(i)cos(47r 0i + 20) . A narrow bandpass filter centered around 2 o passes the pure cosine term in r2, and suppresses the DC component, the (presumably) lowpass v(t), and the upconvert.ed v(t). The output of the bandpass filter is approximately rp(t) BPF r2(t) cos o* + 20 + 0) (10.2) where 0 is the phase shift added by the BPF at frequency 2 o. Since...

## Sampling with Automatic Gain Control

The James Brown canon represents a vast catalogue of recordings -the mother lode of beats - a righteously funky legacy of grooves for us to soak in, sample, and quote. -John Ballon in MustHear Review As foreshadowed in Section 2.8, transmission systems cannot be fully digital because the medium through which the signal propagates is analog. Hence, whether the signal begins as analog (such as voice or music) or whether it begins as digital (such as mpeg, jpeg or wav files) it will be converted...

## Flat Fading A Simple Impairment And A Simple

Unfortunately, a number of the assumptions made in the simulation of the ideal system idsys .m are routinely violated in practice. The designer of a receiver must somehow compensate by improving the receiver. This section presents an impairment (flat fading) for which we have already developed a fix (an AGC). Later sections describe misbehavior due to a wider variety of common impairments that we will spend the rest of the book combating. Flat fading occurs when there are obstacles moving in...

## Wfh wf h24

Thus the spectrum of s(t) consists of two copies of the spectrum of w(t), each shifted in frequency by fo (one up and one down) and each half as large. This is sometimes called the frequency shifting property of the Fourier transform, and sometimes called the modulation property. Figure 2.2 shows how the spectra relate. If w(t) has the magnitude spectrum shown in part (a) (this is shown bandlimited to t and centered at zero Hz or baseband, though it could be elsewhere), then the magnitude...

## Automatic Gain Control

Any receiver is designed to handle signals of a certain average magnitude most effectively. The goal of an AGC is to amplify weak signals and to attenuate strong signals so that they remain (as much as possible) within the normal operating range of the receiver. Typically, the rate at which the gain varies is slow compared to the data rate, though it may be fast by human standards. The power in a received signal depends on many things the strength of the broadcast, the distance from the...

## Carrier Recovery

A man with one watch knows what time it is. A man with two watches is never sure. - Segal's Law Figure 10.1 shows a generic transmitter and receiver pair that emphasize the modulation and corresponding demodulation. Even assuming that the transmission path is ideal (as in Figure 10.1), the signal that arrives at the receiver is a complicated analog waveform that must be downconverted and sampled before the message can be recovered. For the demodulation to be successful, the receiver must be...

## Practical Filtering

Filtering can be viewed as the process of emphasizing or attenuating certain frequencies within a signal. Linear filters are common because they are easy to understand and straightforward to implement. Whether in discrete or continuous time, a linear filter is characterized by its impulse response its output when the input is an impulse. The process of convolution aggregates the impulse responses from all the input instants into a formula for the output. It is hard to visualize the action of...

## T[kSxT[k8

If the r k are too noisy, then the stepsize 1 can be decreased or the length of the average, if present, can be increased , although these will inevitably slow the convergence of the algorithm. The algorithm 12.9 is easy to implement, though it requires samples of the waveform x t at three different points r k S , x r k , and ne possibility is to straightforwardly sample three times. Since sampling is done by hardware, this is a hardware intensive solution. Alternatively, the values can be...

## DJf

f k - fiLPF 27vkTs r kTs - cos 2xkTsf k sm 2xkTsf k . How well does this algorithm work First, observe that the update is multiplied by '2TrkTs this arises from application of the chain rule when taking the derivative of sin 27rfcTs with respect to f k . This factor increases continuously, and acts like a st.epsize that grows over time. Perhaps the easiest way to make any adaptive element fail is to use a st.epsize that, is too large the form of this update ensures that, eventually the...

## Discrete Time And Discrete Frequency

The study of discrete time digital signals and systems parallels that of continuous time analog signals and systems. Many digital processes are fundamentally simpler than their analog counterparts, though there are a few subtleties unique to discrete time implementations. This section begins with a brief overview and comparison, and then proceeds to discuss the DFT, which is the discrete counterpart of the Fourier transform. Just as the impulse function S t plays a key role in defining signals...

## Using An Agc To Combat Fading

One of the impairments encountered in transmission systems is the degradation due to fading, when the strength of the received signal changes in response to changes in the transmission path recall the discussion in Section 4.1.5 on page 75 . This section shows how an AGC can be used to counteract the fading, assuming the rate of the fading is slow, and provided the signal does not disappear completely. Suppose that the input consists of a random sequence undulating slowly up and down in...

## Lpf

Elvis has left the building - Horace Lee Logan, December 15, 1956 5.2 block code, 319, 322 6,4 block code, 325 4-PAM, 14, 40, 157, 204, 207, 210, 213, 228, 253, 284, 315, 331 components, 193 adaptive element, 65-66, 76, 120 Costas phase tracking, 209 CV clock recovery, 250 DD equalization, 279 DD for phase tracking, 213 DMA equalization, 282 LMS equalization, 277 LMS for AGC, 128 output power, 256 PLL for phase tracking, 205 SD for phase tracking, 200 simple AGC, 128 testing, 344 Aeneid, 303...

## Or for JLsa

0 1000 000 3000 4000 5000 8000 7000 8000 3000 1 0000 0 1000 000 3000 4000 5000 8000 7000 8000 3000 1 0000 0 1000 ZKJO 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 ZKJO 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 70110 8000 9000 10W0 FIGURE 6.13 An automatic gain control adjusts the parameter a in the top panel automatically to achieve the desired output power. Perhaps the best way to formally describe how the algorithms work is to plot the performance functions....

## Problems

Explore the behavior of steepest descent by running polyconverge .m with different a Try mu -.01, 0, .0001, .02, .03, .05, 1.0, 10.0. Can mu be too large or too small b Try N 5, 40, 100, 5000. Can N be too large or too small c Try a variety of values of x l . Can x l be too large or too small As an alternative to simulation, observe that the process 6.6 is itself a linear time invariant system, of the general form which is stable as long as a lt 1. For a constant input, the final value...

## Vk

FIGURE 5.5 AM suppressed carrier communication system a the transmitter modulator, b the receiver demodulator. Suppose that the frequency of the modulator is fc but that the frequency at the receiver is fc 7, for some small 7. Similarly, suppose that the phase of the modulator is 0 but that the phase at the receiver is lt f gt . Figure 5.5 b shows this downconverter, which can be described as x t v t cos 27r c j t lt f gt 5.2 where LPF represents a low pass filtering of the demodulated signal x...

## Sm

FIGURE 4.10 The magnitude spectrum of a message signal w t is shown in a . When w t is modulated by a cosine at frequency c, the spectrum of the resulting signal s t w t cos 27rfct lt j gt is shown in b . response scaled by the size of the input and shifted to begin at time A. The complete output is then given by summing over all A. Since there is a continuum of possible values of A, this sum is actually an integral, and the output is x X h t - X dX x t h t . 4.8 This integral defines the...

## R c

Narrow city street, or in a mobile device such as a cell phone the operator might turn a corner and cause a large change in the local geometry of reflections. Fading may also occur when the transmitter and or the receiver are moving. The Doppler effect shifts the frequencies slightly, causing interferences that may slowly change. Such time varying problems cannot be fixed by a single fixed filter, rather, the filter must somehow compensate differently at different times. This is an ideal...

## Simulating The Ideal System

The simulation of the digital communication system in Figure 9.1 divides into two parts just as the figure does. The first part creates the analog transmitted signal, and the second part implements the discrete-time receiver. The message consists of the character string 01234 I wish I were an Oscar Meyer wiener 56789 In order to transmit this important message, it is first translated into the 4-PAM symbol set 1, 3 which is designated m i for i 1,2, , N using the subroutine letters2pam. This can...

## Magnitude spectrum shows the frequency content of the output signal

FIGURE 3.1 The element transforms the input signal x into the output signal y. The action of an element can be thought of in terms of its effect on the signals in time, or via the Fourier transform in terms of its effect on the spectra of the signals. Readers who have studied signal and systems often required in electrical engineering degrees , will recognize that the time domain representation of a signal and its frequency domain representation are related by the Fourier transform, which is...

## Correlation

Suppose there are two signals or sequences. Are they similar, or are they different If one is just shifted in time relative to the other, how can the time shift be determined The approach called correlation shifts one of the sequences in time, and calculates how well they match by multiplying point by point and summing at each shift. When the sum is small then they are not much alike when the sum is large, many terms are similar. Thus correlation is a simple form of pattern matching, which is...

## Timing Recovery

All we have to decide is what to do with the time given us. - Gandalf, in J. R. R. Tolkien's Fellowship of the Ring When the signal arrives at the receiver, it is a complicated analog waveform that must be sampled in order to eventually recover the transmitted message. The timing offset experiments of Section 9.4.5 showed that one kind of stuff that can happen to the received signal is that the samples might inadvertently be taken at inopportune times. The eye becomes closed and the symbols are...

## Digital Filtering And The

Digital filtering is not simply converting from analog to digital filters it is a fundamentally different way of thinking about the topic of signal processing, and many of the ideas and limitations of the analog method have no counterpart in digital form. - R. W. Hamming, Digital Filters, 3rd edition, Prentice Hall 1989. Once the received signal is sampled, the real story of the digital receiver begins. An analog bandpass filter at the front end of the receiver removes extraneous signals for...

## F2 Sketching The Frequency Response From The Ztransform

A complex number a a jb can be drawn in the complex plane as a vector from the origin to the point a,b . Figure F.l gives a graphical illustration of the difference between two complex numbers 3 a, which is equal to the vector drawn from a to 3. The magnitude is the length of this vector, and the angle is measured counterclockwise from the horizontal drawn to the right of a to the direction of 3 a, as shown. FIGURE F.l Graphical calculation of the difference between two complex numbers As with...

## Design Methodology For The M6 Receiver

Before describing the specific design requirements that must be met by a successful M6 receiver, this section makes some generic remarks about a systematic approach to receiver design. There are four generic stages 1. Choose the order in which the basic operations of the receiver occur. 2. Select components and methods that can perform the baste operations in an ideal setting. 3. Select adaptive elements that allow the receiver to continue functioning when there are impairments. 4. Verify that...

## Linear Equalization

The revolution in data communications technology can be dated from the invention of automatic and adaptive channel equalization in the late 1960s. Gitlin, Hayes, and Weinstein, Data Communication Principles, 1992. When all is well in the receiver, there is no interaction between successive symbols each symbol arrives and is decoded independently of all others. But when symbols interact, when the waveform of one symbol corrupts the value of a nearby symbol, then the received signal becomes...

## Telecommunication System

The reason digital radio is so reliable is because it employs a smart receiver. Inside each digital radio receiver there is a tiny Computer a computer capable of sorting through the myriad of reflected and atmospherically distorted transmissions and reconstructing a solid, usable signal for the set to process. from 2 2 03 Telecommunications technologies using electromagnetic transmission surround us television images flicker, radios chatter, cell phones and telephones ring, allowing us to see...

## Kx

FIGURE 10.10 The Costas loop is a phase tracking algorithm based on the performance function 10.13 . Suppose that a 4-PAM transmitted signal r is created as in pulrecsig.m from page 196 with carrier frequency f0 1000. Then the Costas loop phase tracking method 10.15 can be implemented in much the same way that the PLL was implemented in pllconverge .m. costasloop.m costas loop - input rsc from pulrecsig.m initialize estimate vector initialize buffers for LPFs z's contain past fl 1 inputs f 1...

## Power Spectral Density

One way of classifying and measuring signals and systems is by their power or energy , and the amount of power or energy in various frequency regions. This section defines the power spectral density, and shows how it can be used to measure the power in signals, to measure the correlation within a signal, and to talk about the gain of a linear system. In Telecommunication Breakdown, power spectral density is used mainly in Chapter 11 in the discussion of the design of matched filtering. The time...

## Decision device

FIGURE 14.5 Trained Adaptive Linear Equalizer When all goes well, the recursive algorithm 14.27 converges to the vicinity of the block least-squares answer for the particular used in forming the delayed recovery error. As long as 1 is nonzero, if the underlying composition of the received signal changes so that the error increases and the desired equalizer changes, then the f react accordingly. It is this tracking ability that earns it the label adaptive4. The following Matlab code implements...

## Pulse Shaping And Receive Filtering

See first that the design is wise and just that ascertained, pursue it resolutely do not for one repulse forego the purpose that you resolved to effect. - William Shakespeare When the message is digital, it must be converted into an analog signal in order to be transmitted. This conversion is done by the transmit or pulse-shaping filter, which changes each symbol in the digital message into a suitable analog pulse. After transmission, the receive filter assists in recapturing the digital values...

## The Fourier Transform And Its Inverse

By definition, the Fourier transform of a time function w t is which appeared earlier in Equation 2.1 . The Inverse Fourier transform is Observe that the transform is an integral over all time, while the inverse transform is an integral over all frequency the transform converts a signal from time into frequency, while the inverse converts from frequency into time. Because the transform is invertible, it does not create or destroy information. Everything about the time signal w t is contained in...

## Symbols

Eye diagram for the 2T-wide hamming pulse shape FIGURE 11.7 Eye diagrams for T, 2T, 3T, and 5T-wide Hamming pulse shapes show how the sensitivity to noises and timing errors increases as the pulse shape widens. The closed-eye in the bottom plot means that symbol errors are inevitable. shape in the top plot uses ps ones l,M , the Hamming pulse shape in the middle uses ps hamming M , and the bottom plot uses a truncated sine pulse shape ps SRRC L,0,M for L 10. The rectangular pulse is insensitive...

## Example G4

Example G.2 assumes that the parameter ai of the linear filter is fixed, even though the purpose of the adaptation is to allow ai to change in response to the behavior of the signal. Let cr j3 k , k be formed by filtering a signal u k that is independent of 3 k , To be precise, let 3 k ai k and cr j3 k ,k cr ai k , k ao amp amp ai k u k 1 ci2 k u k 2 . Suppose also that the dynamics of ai k are given by ai k ai k 1 nj ai k 1 , k 1 . Then the approximation G.17 yields bi ao k i u k i ai k i u k...