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EECS Advanced Problem Set

This document contains 3 problems from an MIT course on receivers, antennas, and signals: 1) Derive an expression for the root-mean-square temperature difference of a correlation radiometer by analyzing its correlation function and power spectra. 2) Determine the minimum sampling rate and number of shift register stages needed for a digital 1-bit autocorrelation spectrum analyzer given specifications on its frequency resolution, apodization, and aliasing. 3) Prove that a conjugate match maximizes power transfer from a Thevenin source, and determine the exchangeable gain of a negative-resistance load amplifier that uses wave reflection in a transmission line.

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145 views2 pages

EECS Advanced Problem Set

This document contains 3 problems from an MIT course on receivers, antennas, and signals: 1) Derive an expression for the root-mean-square temperature difference of a correlation radiometer by analyzing its correlation function and power spectra. 2) Determine the minimum sampling rate and number of shift register stages needed for a digital 1-bit autocorrelation spectrum analyzer given specifications on its frequency resolution, apodization, and aliasing. 3) Prove that a conjugate match maximizes power transfer from a Thevenin source, and determine the exchangeable gain of a negative-resistance load amplifier that uses wave reflection in a transmission line.

Uploaded by

aree
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Department of Electrical Engineering and Computer Science


Receivers, Antennas, and Signals 6.661
Issued: 02/20/03
Due: 02/27/03

Problem Set No. 3

Problem 3.1
Consider the correlation radiometer of Figure 2.2-11 in the course notes. By
taking the following steps, derive the expression Trms = Teff/(B)0.5 where T2eff = T2A +
2TATR + 2T2R.
a) Find m() in terms of s(0), s(), and n(), assuming E[nanb] = 0 and s and n are
gaussian with zero mean.
b) Find m(f) in terms of s(0), s(f), and n(f).
c) The expression produced in (b) has three parts, i.e. a signal x signal term, a signal
x noise term and a noise x noise term. Graph and fully dimension the power
density spectra corresponding to these three parts (using double-sided spectra in terms
of TA, TR, etc.).
d) Find the DC power and the AC power that emerges in the output vo(t).
e) Find Trms.

-1-

12/22/03

Problem 3.2
A digital 1-bit autocorrelation spectrum analyzer is to be designed with a frequency
resolution of 10 kHz. The autocorrelation function is to be apodized with a weighting
function that reduces spectral resolution by a factor of 1.4 relative to uniform weighting,
while significantly reducing spectral sidelobes. Approximately 1000 independent
spectral intervals are desired at the output, distributed over a 10-MHz band. The effects
of aliasing can be made acceptably low for the given input r.f. filter if the 1-MHz spectral
region just above the desired 1-MHz band is not aliased into that desired band. Note that
both apodization and aliasing increase the number of taps required.
|H(f)|2

1 MHz

a)

f
10 MHz
What is the slowest sampling rate consistent with the foregoing requirements?

b)

How many stages are required for the shift register?

Problem 3.3
a) Prove that a conjugate match ZL = R - jX maximizes the power transferred from a
Thevenin source characterized by the impedance R + jX. This maximum is the
"available power" if R > 0. Assume the source impedance is fixed, and that of the
load is varied to yield the desired maximum.
b) Recalling that the wave reflection coefficient from a load
at the end of a TEM line with characteristic impedance Zo
is = V-/V+ = (ZLn - 1)/(ZLn + 1), what is the
exchangeable gain Ge of a negative-resistance load -RL?
-RL
We define ZLn = ZL/Zo. This amplifier works by reflecting
a signal in a TEM line from a negative resistance load
which is preceded by a three-port circulator (discussed
later in the course; see Eqn. 2.3.37 in the text) that sends
the reflected signal out a different TEM port than the port through which the input
signal entered the amplifier. The negative-resistance load is often a tunnel diode
biased to its negative resistance point.

-2-

12/22/03

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