EECS 554
Homework 8
Due: 11/25
1
Signal Space Representation
1.1 Consider the three waveform fj (t) shown in the following gure.
f1(t)
f2(t)
1/2
f3(t)
1/2
1/2
4
2
1 2 3
4
4
1/2
1/2
1. Show that these waveforms are orthonormal
2. Express the waveform x(t)

EECS 554
Solutions to Homework 5
Due: 10/28
1
Groups
1.1 This problem is added to aid the solution of Problem 2.4. Consider a group
G, and a non-empty subset H of G. Prove that in order to show that H is a subgroup, it is
sufficient to show that (i) addit

EECS 554
Solutions to Homework 6
Due: 11/4
1
Linear Block Codes
1.1 Consider the linear code over Z/7 with check matrix
[
]
1 2 3 4 5 6
1 4 2 2 4 1
Find the minimum distance of this code.
Solution
All columns are non-zero, and no column is a multiple of a

EECS 554
Solutions to Homework 3
Due: 10/7
1
Compression
1.1 In the lecture we stated two equivalent conditions for a prex free (PF) code to be full.
Condition 1: No additional codeword can be added without destroying the prex-free property
Condition 2: N

EECS 554
Solutions to Homework 7
Due 11/13
1
Convolutional Codes
I will post a page from Proakis with the statements of the problems tomorrow.
1.1 Problem 8-1 from Proakis
Solution
1,2,3. The encoder and the state transition diagram associated with this c

EECS 554
Solutions to Homework 4
Due: 10/21
1
Block Codes
1.1 Show that the Hamming distance is indeed a valid distance metric, by proving that it
satisfies the triangular inequality.
Solution
We want to show that for every x, y, z X n we have d(x, z) d(x

EECS 554
Solutions to Homework 1
Due: 9/21
1
Sampling
1.1 Poisson-Sum Formula: Consider a pulse p(t) with Fourier transform P (f ). Consider the periodic
signal generated by placing infinite replicas of this pulse spaced T apart,
i.e., x(t) = k= p(t kT )

EECS 554
Solutions to Homework 2
Due: 9/30
1
Quantization
1.1 A speech signal has a total duration of 10s. It is sampled at the rate of 8 KHz and
then quantized. The signal-to-(quantization) noise ratio is required to be 40 dB. Calculate
the minimum stora

EECS 554
Solutions to Homework 10
Due 12/10
1
Error Probabilities
1.1 Consider the case of communicating over the AWGN channel using the following three
waveforms
cfw_
1, 0 t T
s1 (t) =
0, else
0 t T /2
1,
1, T /2 t T
s2 (t) =
0,
else
s3 (t) = s2 (t)
1.

EECS 554
Solutions to Homework 8
Due: 11/25
1
Signal Space Representation
1.1 Consider the three waveform fj (t) shown in the following gure.
f1(t)
f2(t)
1/2
f3(t)
1/2
2
1/2
4
1 2 3
4
4
1/2
1/2
1. Show that these waveforms are orthonormal
2. Express the w

EECS 554
Homework 10
Due 12/10
1
Error Probabilities
1.1 Consider the case of communicating over the AWGN channel using the following three
waveforms
cfw_
1, 0 t T
s1 (t) =
0, else
0 t T /2
1,
1, T /2 t T
s2 (t) =
0,
else
s3 (t) = s2 (t)
1. Provide tight

EECS 554
Homework 5
Due: 10/28
1
Fields
1.1 Using only the axioms of elds show that for a eld F,
x F ,
x0 = 0x = 0
1.2 For a eld F , show that (1)x = x. Also, show that (1)(1) = 1, where 1 is the
unit element of the eld. Conclude that for positive integer

EECS 554
Homework 9
Due: 12/4
1
General hypothesis-testing problems
1.1 Consider the following binary hypothesis testing problem (assume a-priori probabilities
p0 , p 1 )
H1 : R = S + N
H0 : R = N
where S and N are independent random variables with pdfs
p

EECS 554
Homework 6
Due: 11/4
1
Linear Block Codes
1.1 Consider the linear code over Z/7 with check matrix
1 2 3 4 5 6
1 4 2 2 4 1
Find the minimum distance of this code.
1.2 Find the best (i.e., highest rank) binary linear single-error-correcting code of

EECS 554
Homework 2
Due: 9/30
1
Quantization
1.1 A speech signal has a total duration of 10s. It is sampled at the rate of 8 KHz and
then quantized. The signal-to-(quantization) noise ratio is required to be 40 dB. Calculate
the minimum storage capacity n

EECS 554
Homework 4
Due: 10/21
1
Block Codes
1.1 Show that the Hamming distance is indeed a valid distance metric, by proving that it
satisfies the triangular inequality.
( )
1.2 Using Stirlings approximation to the factorial function, show that nk 2nh(k/

EECS 554
Homework 7
Due 11/13
1
Convolutional Codes
1.1 Problem 8-1 from Proakis
1.2 Problem 8-2 from Proakis
1.3 Problem 8-5 from Proakis
1.4 Problem 8-6 from Proakis
1.5 Problem 8-19 from Proakis
1.6 A generic finite state machine (FSM) is a device with

EECS 554
Homework 1
Due: 9/21
1
Sampling
1.1 Poisson-Sum Formula: Consider a pulse p(t) with Fourier transform P (f ). Consider the periodic
signal generated by placing infinite replicas of this pulse spaced T apart,
i.e., x(t) = k= p(t kT ). Find an exp

EECS 554
Homework 3
Due: 10/7
1
Compression
1.1 In the lecture we stated two equivalent conditions for a prex free (PF) code to be full.
Condition 1: No additional codeword can be added without destroying the prex-free property
Condition 2: No codeword ca

EECS 554
Solutions to Homework 9
Due: 12/4
1
General hypothesis-testing problems
1.1 Consider the following binary hypothesis testing problem (assume a-priori probabilities
p0 , p 1 )
H1 : R = S + N
H0 : R = N
where S and N are independent random variable