NAME ____________________________________________
ES 442 Homework #2
(Spring 2016 – Due February 10, 2016 )
Print out homework and do work on the printed pages.
Textbook: B. P. Lathi & Zhi Ding, Modern Digital and Analog Communication
Systems, 4th edition, Oxford University Press, New york, 2009.
Problem 1 Practice in Watts, dBm and dBW (20 points)
In class you were introduced to the expression of power ratios in terms of dBm and
dBW. Remember that decibels are always used to express ratios of physical quantities
because we can only take the logarithm of unitless numbers (or ratios of physical
quantities because the units cancel out). In electrical engineering it is common practice
to take the logarithm of ratios of power, such as output power to input power (Pout/Pin)
and a power referenced to a standard power like one Watt (P out/1 W). For example, the
power gain of an electronic component expressed in decibels (abbreviated dB) is
P 
P (dB)  10  log10  out  ; expressed in dB
 Pin 
where Pout is the output power and Pin is the input power.
(a) Write an expression for the output power Pout in terms of P(dB) and Pin.
where if PRef is 1 watt, then P is expressed as dBW, and if PRef is 1 mW (0.001 W) P is
expressed as dBm. To get a feel for the magnitudes involved, fill out the table below in
the proper units as specified by the column headings. Use one or two decimal places in
your answers because that is sufficient in most engineering calculations!
Homework 2
(b) If Pin is changed to a reference power, PRef, then we can express a power level, say
Pout in decibels units using
P 
PdB,ref  10  log10  out  ,
 PRe f 
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Power in Watts Expressed in dBW and dBm
Power P in Watts
e.g., 100 watts
Power P in
milliwatts
100,000 mW
Power in dBW
Power in dBm
20 dBW
50 dBm
10 W
1W
0.1 W
0.01 W
0.001 W
2 mW
5 mW
7 mW
0.5 mW
0.2 mW
(c) Why is expressing power levels in dBm or dBW commonly used by electrical
engineers? In other words, what advantage is there to doing this?
Problem 2 Practice using dBm and dBW (20 points)
Homework 2
(a) Various component blocks in communication systems are show below with power
gains or power losses as indicated within the component block. Fill in the blanks for the
four cases given below.
(Table on next page)
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(b) Next, you are given the wireless mobile transmitter-receiver system in the figure
below. Both the transmitter and receiver have antennas and the atmosphere is the
transmission channel in the communication system. Parameter L is the transmission
loss of the channel. The channel is also a source of noise. We are given the following
parameters of the system: (i) The average noise level at the receiver is – 119.5 dBm,
and (ii) We must maintain a signal-to-noise (SNR) ratio at the receiver to properly
operate.
Homework 2
Calculate (1) the average signal strength in dBm at the receiving antenna so that a 30
dB signal-to-noise (SNR) ratio is achieved, and (2) if the propagation loss L is 100 dB,
what is the minimum transmitter power needed to still maintain the 30 dB SNR ratio?
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Problem 3 Channel Capacity and Nyquist Bandwidth (20 points)
(a) Claude Shannon (at Bell Telephone Laboratories) discovered an equation that gives
the highest possible channel capacity of a communication system that can be achieved
in the presence of noise (white Gaussian noise to be specific). The equation is
Shannon channel capacity C  B  log2 (1 SNR ) in bits/second
where B is the channel bandwidth (in Hz) and SNR is signal-to-noise as a power ratio.
Suppose a communication system utilizes a frequency band from 3 MHz to 4 MHz and
requires a SNR (in dB) of at least 24 dB to operate without unacceptable bit errors.
Calculate the maximum channel transmission capacity C as predicted by the Shannon
equation.
[Hint: SNRdB = 10 log10(SNR) when stated in dB, so be careful about mixing the two
logarithmic bases in this problem. The SNR quantity in the Shannon capacity equation
is numerical (not dB) and the logarithm is base 2.]
(b) Now we consider the Nyquist formula which tells us the actual channel capacity as
a function of the number of levels per symbol period. The number of levels per symbol
period for binary is two levels. Other forms of coding can have more than two levels per
symbol. Whereas Shannon’s equation tells the maximum data rate possible in the
presence of noise, Nyquist’s equation tells the data rate C as a function of bandwidth B
and the number of signal levels per symbol achievable (no noise being present).
Nyquist’s equation is
Nyquist channel capacity C  2B  log2 (M ) in bits/second
Homework 2
where M is the number of signal levels per symbol. Calculate the Nyquist data rate
given the same bandwidth as in part (a) above and assume 8 signal levels per symbol.
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Problem 4 Square Law Device (20 points)
A square-law device (such as a MOSFET transistor in saturation) is a nonlinear
component which is useful for signal detection and mixing in communication systems.
Suppose you have a square-law component with an input-output transfer function of
y (t )  A  B  g (t ) 
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where g(t) is the input, A and B are constants and y(t) is the output.
(a) To explore the behavior of this device we let the input signal g(t) be a sinusoidal
tone, that is, g(t) = cos(2ft). Write the expression for y(t) and use the trignometric
identity to convert the cosine-squared term to a first power cosine term. If the input
frequency is f, what is the frequency of the output y(t)?
(b) Unfortunately, no actual device has exactly a square-law characteristic. Some
devices are close, but not perfectly square-law. A better way to express the inputoutput characteristic is with a power series written in the format,
y (t )  A  Bg (t )  C  g (t )   D  g (t )   other terms.
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Homework 2
The first two terms of the expression are linear and can’t generate new frequencies
when driven by a single tone sinusoidal input. In part (a) you found that the third term
does generate an additional frequency. The fourth term, although generally small in
magnitude, is also nonlinear. What frequencies does the cubic term (that is, D[g(t)]3)
generate when driven by g(t) = cos(2ft)?
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Problem 5 RC Low-Pass Filter Problem (20 points)
Filters are a very important part of communication systems. Before we discuss filters in
more detail in class, this problem is assigned to begin thinking about filters; we start with
a very simple filter, namely, the RC low-pass filter. This is illustrated in the figure below:
(a) The transfer function of a filter is defined as the ratio of output signal to input signal
assuming a sinusoidal excitation at the input (i.e., g(f)). The transfer function is
H (f ) 
y (f )
.
g (f )
where g(f) = ej2ft and y(f) = H(f)·ej2ft. Solve for H(f) and find the -3 dB bandwidth (half
power) for this low-pass filter. [Note: This should be review from your ES 400 class.]
Homework 2
(b) Next, we want to explore both the unit step and impulse responses of the RC lowpass filter. A unit step u(t) input is shown in the figure below:
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If we apply the unit step u(t) to the input of the RC low-pass filter what is the output
waveform? Sketch it on the graph below.
(c) The unit impulse response h(t) to a unit impulse function (i.e., Dirac delta function
(t)) is an important function in the characterization of a network (e.g., filter). Determine
the unit impulse response h(t) for this RC low-pass filter and sketch it on the graph
below.
Homework 2
(d) How are the unit step response and the unit impulse response related?
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