ELEN 4810 Homework 1
Analytical Questions
2.23 (a) (1) Suppose that |x[n]| B for all n. Then |y[n]| = | cos(n)x[n]| |x[n]| B. Since
every bounded input x produces a bounded output y, the system is stable. (2) Since y[n] is a
function of x[n] only, the sys

Chapter 3 The szranstorm
3.19. For each of the following pairs of input z-transform X (z) and system function H (z), deter-
mine the ROC for the output z-transform Y (z):
(a)
X() |z|> 1
Z = A l M
1+ﬁz’1 2
H() 1 ll 1
z =——, z >—
1—;1,z_1 4
(b)
X()— 1 |<2
z

ELEN 4810 Final Exam
Monday, December 15, 2014, 9 AM - 12 PM. Two sheets of handwritten notes are allowed. No
electronics of any kind are allowed. Please record your answers in the exam booklet. Raise your
hand if you need additional scratch paper. Good l

ELEN 4810 Midterm Exam
Monday, October 20, 2014, 10:10-11:25 AM. One sheet of handwritten notes is allowed. No electronics of any kind are allowed. Please record your answers in the exam booklet. Raise your hand
if you need additional scratch paper. Good

ELEN 4810 Midterm Exam
Monday, October 20, 2014, 10:10-11:25 AM. One sheet of handwritten notes is allowed. No electronics of any kind are allowed. Please record your answers in the exam booklet. Raise your hand
if you need additional scratch paper. Good

ELEN4810 Digital Signal Processing
The goal of this course is to equip students with basic tools for processing signals on a computer.
We will introduce the basic mathematical ideas necessary to design and analyze discrete time systems, and interface them

ELEN 4810 Homework 6
Analytical Questions
5.22 (a) For h[n] real, any poles which are not pure real occur in complex conjugate pairs. Any
zeros which are not pure real also occur in complex conjugate pairs.
(b) There is a pole at = 0, whose multiplicity i

ELEN 4810 Homework 4
Analytical Questions
4.27 Because we sample at twice the bandlimit, the composite system is LTI when restricted to
bandlimited inputs.
(a) The continuous-time frequency response is
Hc (j)
=
=
Hd (ejT ) | < /T
0
else
ejT /2 ejT /2
T
0

ELEN 4810 Homework 3
Due Wednesday, October 14. Please submit your code for the computational section online
via Courseworks. You can submit your answers to the analytical questions via either
- Hardcopy submission at the beginning of class on Wednesday,

ELEN 4810 Homework 5
Analytical Questions
10.24 (a) x[n] = x(nT ) = ej(3/8)n . Xw [k] will be nonzero at exactly one value whenever (3/8) =
2k/N for some N and integer j. This occurs if and only if 3/16 = k/N . Because 3 does not divide
16, the smallest N

ELEN 4810 Homework 3
Analytical Questions
2.46
(a) We can use the time reversal property of the DTFT:
DT F T
w[n] W (ej ).
(1)
Since
an u[n]
1
1 aej
1
1 aej
DT F T
DT F T
an u[n]
|a| < 1,
(2)
|a| < 1.
(3)
Let v[n] = an u[n]. From the time delay property

ELEN E4810 Digital Signal Processing
Final Exam
Dan Ellis <dpwe@ee.columbia.edu>
Tuesday 2011-12-20 09:00-11:30 (150 min)
This test consists of 4 questions, each with equal weight.
You have two and a half hours (150 minutes) to complete the test.
This tes

Supplementary Notes for ELEN 4810 Lecture 5
Review of Fourier Representations
John Wright
Columbia University
September 23, 2015
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In gen

ELEN E4810 MATLAB Primer
September 11, 2015
This document introduces the basic usage of MATLAB. To obtain MATLAB, Columbia students can go to the
website
http:/portal.seas.columbia.edu/matlab/
and scroll to the bottom of the page.
Arrays and Integer Index

ELEN 4810
Digital Signal Processing
Introduction
John Wright
Electrical Engineering
Columbia University
Welcome to 4810, Digital Signal Processing
Signal: loosely, any information bearing function:
Processing:
Sampling
Denoising / su

Supplementary Notes for ELEN 4810 Lecture 7
Sampling and C/D Conversion
John Wright
Columbia University
October 5, 2014
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In general, you

ELEN 4810 Homework 5
Due Wednesday, November 18. You can submit your answers to the analytical questions via
either
- Hardcopy submission at the beginning of class on Wednesday, November 18, or
- Electronic submission (in pdf form) on Courseworks. Please

ELEN 4810 Homework 3
Due Wednesday, October 14. Please submit your code for the computational section online
via Courseworks. You can submit your answers to the analytical questions via either
- Hardcopy submission at the beginning of class on Wednesday,

Supplementary Notes for ELEN 4810 Lecture 8
The Discrete Fourier Transform
John Wright
Columbia University
October 12, 2015
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at
completeness. In general,

ELEN 4810 Homework 2
Due Wednesday, September 24. Please submit your code for the computational section online
via Courseworks. You can submit your answers to the analytical question, as well as the output of
your code via one of three means:
- Hardcopy s

Supplementary Notes for ELEN 4810 Lecture 4
More Linear Time Invariant Systems
John Wright
Columbia University
September 21, 2015
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at
completeness. In ge

Supplementary Notes for ELEN 4810 Lecture 6
The Discrete Time Fourier Transform
John Wright
Columbia University
September 28, 2015
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In g

Supplementary Notes for ELEN 4810 Lecture 10
Windowing and the Short-Time Fourier Transform
John Wright
Columbia University
October 21, 2015
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at complete

Supplementary Notes for ELEN 4810 Lecture 16
FIR Filter Design
John Wright
Columbia University
December 2, 2015
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In general, you can nd

Supplementary Notes for ELEN 4810 Lecture 17
Innite Impulse Response (IIR) Filter Design
John Wright
Columbia University
Decmeber 11, 2015
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completene

ELEN 4810 Final Exam
Monday, December 15, 2014, 9 AM - 12 PM. Two sheets of handwritten notes are allowed. No
electronics of any kind are allowed. Please record your answers in the exam booklet. Raise your
hand if you need additional scratch paper. Good l

Chapter4
246
Sampling of Continuous-Time Signals
4.27. The system shown in Figure P4.27 is intended to approximate a differentia tor for bandlimited continuous-time input wavefonns.
~-~
'
I
I Yc( t)
D/C
C!D
I
I
I
I
I
I
I
I
T
T
~-~
Figure P4.27
The contin

ELEN 4810 Homework 3
Due Wednesday, October 14. Please submit your code for the computational section online
via Courseworks. You can submit your answers to the analytical questions via either
- Hardcopy submission at the beginning of class on Wednesday,

ELEN 4810 Homework 4
Due Wednesday, October 21. You can submit your answers to the analytical questions via
either
- Hardcopy submission at the beginning of class on Wednesday, October 21, or
- Electronic submission (in pdf form) on Courseworks. Please na

ELEN 4810 Homework 6
Due Wednesday, December 2. You can submit your answers to the analytical questions via
either
- Hardcopy submission at the beginning of class on Wednesday, December 2, or
- Electronic submission (in pdf form) on Courseworks. Please na

Supplementary Notes for ELEN 4810 Lecture 11
The Z-transform
John Wright
Columbia University
November 9, 2014
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In general, you can find

Supplementary Notes for ELEN 4810 Lecture 14
Minimum Phase All-Pass Decomposition
John Wright
Columbia University
November 23, 2015
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In

In the interest of clarity, here is a much simpler way. Given the sampled signal x[n], with DTFT
X(ej ), the reconstructed signal in time domain is
xr (t) =
X
x[n]hr (t nT 0 ),
(0.1)
n=
where hr (t) is the ideal (sinc) reconstruction filter. In frequency

Supplementary Notes for ELEN 4810 Lecture 7
Sampling and C/D Conversion
John Wright
Columbia University
October 3, 2016
Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In general, you

ELEN 4810
Digital Signal Processing
Introduction
John Wright
Electrical Engineering
Columbia University
Welcome to 4810, Digital Signal Processing
Signal: loosely, any information bearing function:
Processing:
Sampling
Denoising / su