Chapter 6 Problems 1.3 2003/06/02 2003/09/05 16:29:26.385 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu rectangular jam orthogonal amplitude limited diversity signaling Doppler synchronization phase tracking N-dimensional rectangular signal sets have signals located at the vertices of an N -dimensional hypercube. Thus there are K 2 N signals in the signal set. For N 2 , here is the signal constealltion

The signals chosen for this signal set are sinusoids of duration T ( s i t ± 2 E T 2 f i t , 0 t T ). The total bandwidth used by the entire signal set is to be no more than W.

Find the maximum dimension of a rectangular set that fulfills these design criteria.

What is the resulting probability of error for the optimum receiver when this signal set is used over an additive white Gaussian noise channel?

The following signal set is used to transmit one of three equally likely messages. s 0 t 2 E T 2 f c t s 1 t E T 2 f c t 4 s 2 t 2 E T 2 f c t where the observation interval is 0 T and the frequency f c is harmonic with this interval.

Sketch the signal constellation, labeling your sketch accurately.

Find the minimum probability of error receiver for this signal set.

Find the probability of error that results when your receiver is used.

A ternary signal set consists of three (equally likely) signals. A clever engineer chooses the following signal set defined over 0 T . s 0 t 2 E 2 f 0 t s 1 t 0 s 2 t 2 E 2 f 0 t The channel adds white Gaussian noise to the transmitted signal.

Find the minimum probability of error receiver for this signal set.

What is the resulting probability of error?

Now assume that the received signal energy E is unknown. How would your receiver be modified to take this uncertainty into account?

Binary pulse-position modulation (PPM) consists of encoding a bit by the position in time of a given pulse waveform p t within the bit interval. More precisely, t 0 t T R t p t N t t 0 t T R t p t τ N t The pulse p t is a square pulse having a duration less than the bit interval duration T. The amplitude of p t cannot exceed the value of A . The delayed pulse p t τ is not allowed to spillover into the next bit interval. The hypotheses (i.e., bits) are equally likely.

Find the optimal signal set (i.e., choose values for τ, the amplitude, and the duration of p t ) when N t is white Gaussian noise.

What is the optimal receiver for this signal set?

What is the average probability of error obtained with this receiver?

The depicted signal set is used to transmit equally likely binary messages.

Assume the channel adds white Gaussian noise of spectral height N 0 2 . What is the minimum attainable probability of error that any receiver can achieve?

An engineer decides that performance obtained above is adequate for his application. She decides to adopt this signal set for transmission over the channel shown in .

The impulse response of the channel filter is h t δ t δ t 1 The channel noise n t is Gaussian with power density spectrum 𝒮 n f 1 . Draw and label accurately the minimum probability of error receiver.

What probability of error does your receiver achieve?

A second engineer also wants to use this signal set (it must be a good one!). His channel can be modeled as

If the characteristics of n t are the same as previously, what is the minimum probability of error that can be achieved by any receiver?

An engineer proposes to use the following binary signal set over an additive, white, Gaussian noise channel. s 0 t E 0 t 2 0 s 1 t E 0 t 1 0 The signals are equally likely to be sent.

Sketch the signal constealltion for this signal set and indicate the location of the optimum decision boundary.

Calculate the average probability of error for the optimum receiver.

The engineer wonders whether his designed signal set/optimum receiver will be sensitive to amplitude variations of the receiver signals. Are his concerns valid? If so modify the signal set without changing the waveforms (i.e., the modified signals should be scaled versions of the original signals) so that the modified set won't be sensitive to amplitude variations. If not, demonstrate that the original set can be used in this situation.

In desperate need of help, the Rice football coaching staff has installed a digital communications system to relay messages from the press box to the field. A student designs the following binary signal set. s 0 t 2 f 0 2 f 0 t s 1 t 2 f 0 2 f 0 t The duration of each transmission interval is 1 f 0 and the frequency f 0 is 1 MHz. The communications is modeled as an additive, white Gaussian noise channel with unit spectral amplitude. Assume the signals are equally likely.

What is the minimum probability of error that any receiver can achieve when this presumably well-designed signal set is used?

In College Station, the Aggies decide to jam this system by sending a constant amplitude cosine wave. To the receiver, the channel adds to the transmitted signal the sinusoid 2 f 0 2 f 0 t as well as noise having the same characteristics as described above. Find the minimum probability of error receiver and its resulting performance.

Not to be outdone, clever engineers from the MOB decide to jam the well-designed Rice system by transmitting a cosine wave whose amplitude alternates between 2 f 0 and 2 f 0 . This jamming signal has a fixed amplitude over each transmission interval, and the amplitude is equally likely to be positive or negative. The sign of the jamming signal during a particular transmission interval is statistically independent of the sign in any other interval. Assuming the noise remains the same, what is the minimum probability of error receiver and what is the resulting performance?

A binary signal set is to be designed for communication over a Poisson channel. Let λ 0 t and λ 1 t denote the signals (i.e., intensities) comprising the signal set. These signals are constrained so that t 0 T λ 0 t t 0 T λ 1 t where T is the duration of the bit interval. These signals are equally likely to occur.

Find the optimal signal set for this communications system.The answer is not unique. Indicate why this is so and find one of the optimal sets.

Compute the probability of error that results when the optimal receiver is used.

A Poisson process channel is used to communicate one of two equally likely signals by using the signals λ 0 t and λ 1 t as the intensities of the process over 0 T .

Find the minimum probability of error receiver.

Contrast the computations required by this receiver with those you would expect when the channel adds white Gaussian noise to the transmitted signal. In particular, can the receiver for the Poisson channel be put in the form of a matched filter? Show how or how not.

A digital transmitter and receiver pair agree to use the binary signal set shown in

Assuming the channel adds white Gaussian noise to the transmitted signal, find the minimum probability of error receiver. Assume that the signals are equally likely. Sketch the signal constellation and determine the probability of error attained by your receiver. The receiver and transmitter agree (for some reason) to use an M-ary signal set consisting of signals of the form: i i 1 … M s i t i 1 2 E 0 T 2 f 0 t The channel adds white Gaussian noise to the transmitted signal.

Find the minimum probability of error receiver.

Sketch the signal constellation and compute the probability of error.

What is the average transmitted energy? What translation of the signal constellation has the minimum transmitted energy but maintains the probability of error?

Let s i t , i 1 … M denote the signals comprising a transmitter's signal set. The set is said to be an orthogonal signal set if the signals are pairwise orthogonal: s i s j 0 , i j . Usually, the signals in an orthogonal signal set have equal energies. i i 1 … M s i 2 E Under these conditions and assuming the members of the set are equally likely, solve the following problems.

Sketch a block diagram of the minimum probability of error receiver when the channel adds white Gaussian noise to the transmitted signal. Do not assume M 2 .

Now assume the channel is a fading channel. The amplitude of the transmitted signal in each bit interval is scaled by a random variable having a density symmetric about the origin. How do you modify the receiver of the previous part to accommodate this channel?

Assume M 2 and the channel adds colored Gaussian noise to the transmitted signal. Find the orthogonal signal set having the smallest probability of error. Assume that singular detection is NOT possible.

The Cheap and Dirty Communications Company has designed yet another digital communications system. The binary signal set consists of a sinusoid having duration T and frequency f 0 ( = n T ) and its negative. These signals are equally likely. The channel adds white Gaussian noise of spectral height N 0 2 . An average bit error rate of at least 10 -3 is required. The transmitter is amplitude-limited: sinusoids must be less than 2 A 0 in amplitude to be transmitted.

What is the minimum probability of error receiver?

What is the maximum data rate (bits/second) that can be transmitted through the channel?

The designer can save some expense (the motto and Cheap and Dirty) by simplifying the receiver. Instead of using sinusoids in the receiver, square waves of the same frequency and phase are used to reduce hardware costs. Analyze the effect of this less expensive receiver on the performance of the communications system.

A binary signal set is used over an additive white Gaussian noise channel. The signals comprising the signal set are equally likely and are given by s 0 t A 2 f 0 t and s 1 t A 2 f 1 t where f 0 n 0 T and f 1 n 1 T with T denoting the duration of a bit interval and n 0 , n 1 denoting distinct integers. The amplitude of each signal is unknown at the receiver.

Assuming that the amplitude is positive, find the minimum probability of error receiver.

The same signal set is to be used in a sequential detection system. With specified false-alarm and detection probabilities, find the optimum sequential detection system assuming that the amplitude is unknown.Be careful in your derivation; the result is not obvious.

What is the expected number of transmissions required to meet the criteria?

Suppose the transmission signal set contains four signals. s 0 t A 2 f 0 t s 1 t A 2 f 1 t s 2 t A 2 f 0 t s 3 t A 2 f 1 t where f 0 n T and f 1 m T , n m and T is the transmission interval. Each signal is equally likely. The channel corrupts transmissions by adding white Gaussian noise.

Sketch a block diagram of the minimum probability of error receiver. Sketch signal space and indicate the decision regions.

Compute the probability of error.

We have assumed that the time-origin of data transmission is known by the receiver. In practical systems, this time reference must be derived from the transmission of a known bit sequence-the preamble-and then used by the receiver for determining succeeding data transmissions. Assume the preamble consists of a sequence of identical bits, and that the signal representing this bit is a square-wave signal, equaling E T during the first half of the bit interval and E T during the second half. The channel adds white Gaussian noise to the transmissions. Derive a method for estimating the difference between the transmitter's and receiver's time reference from the observations taken over a single bit interval. Practical solutions should be as simple as possible: Determine as simple a receiver as you can. One potential problem in digital communication is interference from other digital transmitters as well as from the usual channel noise. Assume transmitter A is using signal set A and transmitter B signal set B.

Assume the signals in each set are equally likely. The receiver trying to pay attention to transmitter A receives the signal r t s A t a s B t N t where a is an unknown, fixed, positive constant and N t is white Gaussian noise. Assume that the transmitters A and B are synchronized so that the bit intervals coincide. The signals sent by each transmitter are statistically independent.

Determine the minimum probability of error receiver for the reception of transmitter A's signals in this situation.

What is the resulting probability of error?

Demonstrate that signals from transmitter B can also be received without knowing the value of a.

In realistic radar problems, the number of airplanes within the radar's operating range is unknown. Assume that radar return from the m th airplane equals s t τ m , where s · represents a known signal and τ m a delay unknown save for the fact it lies in the observation interval 0 T . The observations consist of a sum of M such returns, where M is unknown (and could be zero). When more than one return is present, they do not overlap each other (else the airplanes would be very close together). The measured returns are contaminated by additive white Gaussian noise having known spectral height.

For the moment, assume M is some known number. What receiver maximizes the probability of detecting the presence of all airplanes under a false-alarm probability constraint?

When M is totally unknown, what is the optimum receiver?

An engineer is designing a digital communication system to be used in situations where the amplitude of the received signals is unknown. A binary signal set is desired where the symbols to be transmitted are equally likely. The channel adds an unknown quantity of Gaussian noise to the transmitted signal.

Assuming the noise is white, design a binary signal set and the accompanying receiver which has the best possible performance.

The engineer finds the situation to be worse than expected. The noise is found to not be white, but the power density spectrum is unknown. However he/she decides to use the receiver designed for the white Gaussian noise channel. Find an expression for the probability of error when the suboptimum receiver is used.

Under what conditions will this receiver be optimum?

The signal s t , 0 t T , may be transmitted with the probability 1 2 . If s t is not transmitted, nothing will be sent. The channel is characterized as adding Gaussian noise having covariance function K N t u 1 t u t u 1 0 to the transmitted signal. Assume that s t is twice differentiable and that T 1 .

What is the optimum receiver? Find the function g t with which the receiver matches the received signal.

Find a general expression for the probability of error in terms of s t and g t .

What signal(s) s t will make this detection problem singular?

A well-performing digital communication system (obviously designed by a Rice engineer) has been in operation for several months. It employs anitpodal signaling for transmitting a bit stream over a colored Gaussian noise channel (having known covariance function). Each signal takes T seconds to transmit. The engineer's boss (an Aggie) thinks that he can increase the rate at which digital information is transmitted over the channel by simultaneously transmitting a second binary datastream synchronously with the first (they share the same bit interval). The Rice engineer is charged with the risk of designing the new system; the only restriction is that the original signal set cannot be changed.

What second signal set should she choose? Justify your answer.

What receiver decodes the two datastreams simultaneously?

How, if at all, has the performance of the original system been changed by the presence of the second datastream? Discuss the viability of the proposed system.

An anitpodal signal set is to be used over a particular multipath channel. The impulse response of this channel is given by h t δ t a δ t Δ where Δ is a known delay, but a is an unknown amplitude. Because of the multipath, the signals are adjusted so that the signal and its delayed version do not overlap in time and so that the entire signal and its delayed version are contained within a bit interval. The channel also adds white Gaussian noise to the output of the multipath portion of the channel.

Find an optimum receiver for this channel.

Find an expression for the false-alarm probability of your receiver.

If the channel added colored instead of white noise, how would your receiver change?

Diversity signaling is the transmission of the same message over N distinct channels simultaneously to a receiver. If the statistical characteristics of each channel are independent of each other, potentially an improvement in performance can be obtained. Assume that an on-off signalling scheme is used over each of N Rayleigh channels. The carrier frequencies and bandwidths of the N signal sets are chosen such that little overlap occurs. The received signals are of the form: t i 0 t T i 1 … N R i t N i t t i 0 t T i 1 … N R i t A i 2 m i t 2 f i θ i N i t The amplitudes A i are statistically independent Rayleigh random variables having variance σ 2 and the phases θ i are statistically independent uniform random variables, distributed over . The additive noise N i t is white Gaussian noise having a spectral height N 0 2 ; the noise in one channel is independent of the noise in the other channels. The energy of each of the signals m i t is E N .

Find the optimum receiver which can view the output of only one channel.

Find the optimum receiver which can view the outputs of all of the channels.

Find an expression for the probability of error when the hypotheses are equally likely for each receiver.

Compare the results of the first receiver (taking N 1 ) and the second receiver when N 1 . Does diversity signalling result in improved performance?

Modern management styles tend to want decisions to be made locally (by people at the scene) rather than by "the boss." While this approach might be considered more democratic, we should understand how to make decisions under such organizational constraints and what the performance might be. Let three "local" systems separately make observations. Each local system's observations are identically distributed and statistically independent of the others, and based on the observations, each system decides which of two hypotheses best applies. The judgments are relayed to the central manager who must make the final decision. Assume the local observations consist either of white Gaussian noise or of a signal having energy E to which the same white Gaussian noise has been added. The signal energy is the same at each local system. Each local decision system must meet a performance standard on the probability it declares the presence of a signal when none is present.

What decision rule should each local system use?

Assuming the observation models are equally likely, how would the central management make its decision so as to minimize the probability of error?

Is this decentralized decision system optimal (i.e., the probability of error for the final decision is minimized)? If so, demonstrate optimality; if not, find the optimal system.

A channel is found to be dispersive and to add colored Gaussian noise to the dispersed transmitted signal.

Find the minimum probability of error receiver and the equations governing it.

Suppose we have the special case where the additive noise N t is well described as passing white Gaussian noise through h CH t τ , the same filter that the signal passes through. Now what is the structure of the receiver?

We wish to design a binary signal set optimizing the performance of a digital communications system that must use a dispersive channel, which filters the transmitted signal then adds noise. The transmitted signals are constrained to have duration T and to have an energy expended over this interval no larger than E. The impulse response of the channel is given by: h CH t α α t u t The additive noise N t is described as being white and Gaussian. The receiver is constrained to observe the received waveform over an interval of duration T. Assume that the intersymbol interference is negligible. Find the average error rate of your receiver. What is the equivalent loss of signal energy due to the filtering characteristics of the channel? Consider a channel which contains a dispersive filter affecting the transmitted signal and adds white Gaussian noise to the output of the filter. Let h CH t τ denote the impulse response of this filter.

Show that the quantity Q CH t u is defined by Q CH t u α h CH α t h CH α u is a positive-definite function for all h CH t τ .

Show how to find the optimal orthogonal signal set for this channel.

How does the performance of this channel compare with that of the optimal anitpodal signal set?

Let a transmitter use an anitpodal signal set defined by the signal s 0 t . t 0 t T s 0 t E T The channel is characterized by a dispersive filter having the impulse response h CH t . h CH t α α t u t The output of this filter is then contaminated by additive, white Gaussian noise. The receiver observes the resulting signal over an interval of duration T. The receiver is a matched-filter receiver which is matched to the transmitted signal.

What is an expression for the output of the matched filter at the end of each observation interval?

Assuming that only the effects of the preceding bit interval are significant, find an expression for the average probability of error that results if the receiver ignores intersymbol interference.

Draw a block diagram of a decision-feedback receiver which can be used to reduce the intersymbol interference. Evaluate the resulting performance of this receiver and compare to the result of not using this receiver (previous part).

A more reasectionic model of a radar return is that the received signal is of the form R t A 2 m t τ 0 2 f c t τ 0 θ N t where the duration of the signal m t is assumed to be much less than the observation interval 0 T . Assume that edge effects are negligible. The quantities A , θ , and τ 0 are unknown. The noise N t is white and Gaussian.

Find a receiver for detecting the presence or absence of this return.

Now assumed that the frequency f c is unknown, but lies in the range f l f h . This situation can occur when the target is moving away from the radar, thereby inducing the Doppler effect on the return. What receiver can operate in this environment?

One method of communicating over a channel in which its parameters vary slowly compared with a bit interval is to precede the information-bearing portion of the bit interval with a known probe signal. This signal can then be used to provide some information about the channel which can be used to aid in the detection problem. Assume a modulated signal set is used over a random phase channel. The baseband probe signal m 0 t is always transmitted over the first half of the bit interval. The baseband message signal ± m 1 t is used to transmit equally likely binary information in the second half. The received signal is of the form R t 2 m 0 t m 1 t 2 f c t θ N t R t 2 m 0 t m 1 t 2 f c t θ N t where N t is white Gaussian noise and E 0 and E 1 , the energies of the probe and message signals, are equal.

Assume the phase θ is a known constant. Show that the optimum receiver ignores the probe portion of the received signal.

Now assume that θ is a random variable uniformly distributed over the interval . Find the minimum probability of error receiver.

Show that this receiver can be put in the form of a phase discriminatory where the phases of the received signal in the probe and message portions of the received signal are compared. The discriminatory announces ℳ 1 if the phase difference is greater than 90° in magnitude and ℳ 0 otherwise.

Find the probability of error of the optimum receiver.

How does the performance of this signal set compare with that when no probe signal is used and the same message signal set is used?

One of the most common problems in digital communication is synchronization of the receiver to the sequence of transmitted signals. To help the receiver achieve synchronization, the transmitter sends a sequence known to the receiver. For example, the sequence alternating between s 0 and s 1 is often used. Assume that the binary signal set consists of s 0 t 0 and s 1 t E T , 0 t T . The time origin of the receiver is assumed to be initially out of synchronization with that of the transmitter by an amount τ as shown. White Gaussian noise is added to the signal by the channel.

If the receiver does not attempt to compensate for the lack of synchronization, what is the resulting probability of error?

One of the important aspects of the solution to the synchronization problem is to determine whether the offset τ between the receiver's and the transmitter's time origin is less than half of the duration of a bit interval or not. An engineer suggests considering two successive bit intervals and determining which interval contains more non-zero signal. Describe a procedure which will make this determination in the best possible way. What are the error characteristics of your approach?

Once this determination is made, the engineer claims that the value of τ can be determined while taking no longer than two successive bit intervals. What might his technique be?

In reasectionic radar problems, the return signal s t is not only delayed by an unknown amount τ but is also attenuated by an amount dependent on the range of the target (hence the delay τ). Assume that the delay ranges between τ min and τ max and that the signal model for the radar return is s t k τ s 0 t τ where s 0 t has duration T (which is much less than τ max τ min ) and energy E. The probability of the presence of a return is unknown. Design a receiver which can determine the presence or absence of a return in white Gaussian noise over the observation interval τ min τ max T . Explicitly indicate the details of your receiver, including the values of constants and threshold values. A modulated anitpodal signal set is used over a channel which changes the phase of the transmitted signal by 90° or leaves the phase unchanged. This phase shift changes randomly from bit-to-bit and is equally likely to change the phase or not. The transmitted signals are equally likely to occur.

Find the optimum receiver for this channel.

Calculate the resulting probablilty of error for your receiver.

The usual random-phase receiver does not assume any information about the phase of the previous bit intervals. In this problem, we explore a phase-tracking receiver. Let the signal set defined over 0 T be s 0 t 2 E T 2 f 0 t s 1 t 0 The channel provides a random phase shift and then adds white Gaussian noise.

Let θ ̂ k denote the receiver's guess of the phase of the carrier in the k th under bit interval. The actual phase of the carrier is denoted by θ k . Let ε k denote the value of the output of a matched filter which uses this phase guess. This output will then be used to update the phase guess for the next bit interval. To be maximally sensitive to the difference φ k θ ̂ k θ k , should the receiver match to the in-phase or quadrature component of the received signal?

Assuming the receiver uses the matched filter found in the previous part, what is the expected value and variance of ε k ?

We wish to update this guess for each bit interval according to θ ̂ k + 1 θ ̂ k α ε k where α is a design constant. Write this difference equation in terms of φ k and linearize it (i.e., assume small φ k ). What is the steady-state variance of φ k ? What is the bandwidth of the difference equation? Discuss the tradeoff between the variance of φ k and the bandwidth of φ k . Why are we concerned about bandwidth?

Suppose a receiver incorporates this phase-tracking procedure into its structure. Sketch a block diagram of this receiver.

What is the resulting probability of error of this receiver when a fixed phase error is assumed?

Assume the phase error of the previous part is twice the standard deviation of φ k . Let E N 0 4 and the transmission rate be 2400 bits/sec. What is the resulting degradation in the probability of error due to a 0.1 radian guessing error? What is the value of α which gives this error? What is the resulting bandwidth?

Often the received signal is distorted in the communications process in nonlinear, unpredictable ways. Distortion may be present in the transmitter, the channel, and in the front end of the receiver. Assume s 0 t is the intended signal. The received signal is assumed to be of the form s t s 0 t s δ t , where s 0 2 E and s δ t is the distortion signal which is known only to the extent that it is limited in energy: s δ 2 ε 2 ( E ε 2 ). An anitpodal signal set of equally likely components is used and the channel, in addition to contributing (possibly) to the distortion, adds white Gaussian noise to the transmitted signal.

Assuming that no distortion is present, what is the signal constellation and what is the minimum probability of error that can be achieved with any receiver?

Assuming that the receiver achieving the best possible performance is used, what is the worse-case distortion signal that could be received?

What receiver can best cope with this worse-case situation? This receiver would therefore minimize the worse possible performance in the presence of this distortion.

In realistic radar problems, the number of airplanes within the radar's operating range is unknown. Assume that radar return from the m th airplane equals s t τ m , where s · represents a known signal and τ m a delay unknown save for the fact it lies in the observation interval 0 T . The observations consist of a sum of M such returns, where M is unknown (and could be zero). When more than one return is present, they do not overlap each other (else the airplanes would be very close together). The measured returns are contaminated by additive white Gaussian noise having known spectral height.

For the moment, assume that M is some known number. What receiver maximizes the probability of detecting the presence of all airplanes under a false-alarm probability constraint?

When M is totally unknown, what is the optimum receiver?