Van Trees Vol1 A Mathematical Look At


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A presentation which summarizes
the Mathematics of possibly the
most celebrated book in communication theory

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  • Van Trees Vol1 A Mathematical Look At

    1. 1. A Mathematical view of Van Trees- Volume I Tim Mazumdar Dr. Govind R. Kadambi Feb - 2010
    2. 2. Source /Transmitter/Channel <ul><li>Source equation </li></ul><ul><li>Due to phase shifts in the TX oscillator this can be written as , </li></ul>
    3. 3. Radar and Sonar Problem VT 7 through 9 <ul><li>Detection problem – detect s1(t) in the presence of noise. </li></ul><ul><li>RADAR Problem – Detect presence or absence of target in case </li></ul><ul><li>Of received signal has noise. We first write the transmitted signal </li></ul><ul><li>the received signal and look at probability of detection </li></ul><ul><li>The transmitted signal can be written as, </li></ul><ul><li>Vtau is due to attenuation of the transmitted signal and phi due to phase Shift of the reflected signal from the target. </li></ul><ul><li>Is the round trip travel time to target. That is source to target and </li></ul><ul><li>back to source. </li></ul><ul><li>The attneuation factor, phase shift and round trip time are unknown </li></ul><ul><li>In a general sense. </li></ul>
    4. 4. Third class of communication systems – Passive SONAR and Cog Radio <ul><li>In the Passive case its either signal + noise or only noise </li></ul><ul><li>r(t) = s(t) + n(t) – signal present </li></ul><ul><li>r(t) = n(t) –Signal absent. </li></ul><ul><li>The task is to look at the statistical nature of the two signals </li></ul><ul><li>And try to distinguish between them </li></ul><ul><li>This is called the binary detection problem. </li></ul><ul><li>In the generalized case M signals will be there and it is called </li></ul><ul><li>A M-ary detection problem. </li></ul>
    5. 5. The 3 classes of Detection problems in Communication
    6. 6. Pulse amplitude modulation Pulse frequency modulation In the first case the value of the amplitude of TX is dependent on the signal supplied By the modulator. In the second case the value of the frequency of TX is modulated on the signal supplied By the modulator. What does the receiver do when it receives such a PAM or PFM signal
    7. 7. The received signal for PAM or PFM <ul><li>The received signal = PAM + noise </li></ul><ul><li>First level of estimation problem in hierarchy – KNOWN SIGNAL in NOISE </li></ul><ul><li>If noise is absent and an inverse function can be defined then the An is known </li></ul><ul><li>From above just by studies r(t) = s(t,An) </li></ul><ul><li>KNOWN SIGNAL WITH UNKNOWN PARAMETERS IN NOISE. </li></ul><ul><li>A CLASSIC EXAMPLE IS RADAR SIGNALS WHERE </li></ul><ul><li>1). Target velocity is not known – contributes to Doppler shift in received signal. </li></ul><ul><li>2). Target distance is not known – contributes to received window in Doppler </li></ul><ul><li>Sifter signal. </li></ul><ul><li>We can add two more to this </li></ul><ul><li>3). Received signal amplitude is not known An is a random variate. </li></ul><ul><li>4). Phase shift of the received signal is an unknown random variate. </li></ul><ul><li>The equation which captures this difficult problem is as follows </li></ul>
    8. 8. Amplitude Modulated and Frequency modulated signals <ul><li>An amplitude modulated signal is such that its amplitude varies in time while its frequency keeps constant. </li></ul>The Problem of estimation of a unknown channel <ul><li>In this case the output is a discrete time convolution of the input sequence x(n) and the channel transfer function h(k) . </li></ul>
    9. 9. Simple diagram of Continuous Channel Model <ul><li>The received tries to analyze r and obtain h(t). Also called CHANNEL ESTIMATION . </li></ul><ul><li>Another associated problem in Modulation theory is the estimation of the phase shift in the channel. Formally this can be written as follows, </li></ul><ul><li>This is AM without Phase synchronization. </li></ul>
    10. 10. Approach to Detection and Estimation theory The approach must estimate unknown parameter like phase shift, amplitude all in presence Of noise. Even without noise these parameters are unknown. All of Detection Theory and Estimation theory is statistical in approach. Let there is a receiver of signals with signal and noise at its input. R(t) = s(t) + n(t) between 0 and T = 0 otherwise The energy due to the signal is computed as the integral, Define SNR = Energy of Deterministic signal/Energy is defined as,
    11. 11. Moving E into the integral Moving the E operator inside the autocorrelation integrals is not permissible under normal circumstances The double integral can be reduced to a single integral using Fubini’s theorem.
    12. 12. The 3 ideas of Van Trees wrt. Stochastic problems <ul><li>His 3 ideas are crucial because that is how stochastic problems in communication and </li></ul><ul><li>Control can be addressed in a unified framework. </li></ul>
    13. 13. What are the problems wrt. Non linear systems <ul><li>Convolution integral does not apply to non linear systems. </li></ul><ul><li>Example of simple quadratic function non linear system </li></ul><ul><li>Example of memoryless non linear channel with added noise </li></ul><ul><li>The 3 coefficients a0, a1 and a2 must be chosen . Criterion is some form </li></ul><ul><li>for expectation maximization or minimization. This is the general class of communication </li></ul><ul><li>Problems. </li></ul>
    14. 14. Summary <ul><li>Stochastic techniques like moments, characteristic functions , </li></ul><ul><li>Stochastic maximization are essential to solve communication problems. </li></ul><ul><li>They require a knowledge of calculus , Hibert spaces, Complex variable theory </li></ul><ul><li>And state variables </li></ul>