Signal Processing - November 2016 - 140

close to the pseudoinverse transform,
however, implementing the true adjoint
is just a simple matter of handling the
signal extension operation.
Another interesting adjoint appears
in a blind channel estimation optimization problem. The convolution of two
variables, which is a nonlinear operation, can be written as a matrix-vector
product in two ways. With this viewpoint, deriving the gradient becomes
simple and a fast implementation is
immediate. For an impulsive source
sent through a simulated underwater
acoustic channel, the blind channel estimation problem is able to accurately
recover important details of the acoustic
channel response.

x_true

1
0
-1

500

1,000

1,500

2,000

2,500

3,000

2,500

3,000

x_est = conv(h_est, s_est)

1
0
-1

500

1,000

1,500
Time Index

2,000

(a)
h_true

2
0
-2

∆t = 0.057s
0

100

300

400

500

600

700

800

h_est

0.2

Supplementary material

∆t = 0.0575s

0
-0.2

200

100

200

300

400
500
Index

600

700

800

1,200

1,400

1,600

This article has supplementary downloadable material available at https://
g i t h u b. c o m / j a m e s f o l b e r t h / i e e e _
adjoints, provided by the authors. The
material includes MATLAB implementations of the adjoint operators discussed in this lecture note. Contact
james.folberth@colorado.edu for further questions about this work.

(b)
s_true

0.5
0
-0.5

200

400

600

800

1,000

Authors

s_est

5
0
-5

200

400

600

800
1,000
Index

1,200

1,400

1,600

(c)

Figure 4. The blind channel estimation via problem (6) for a simulated underwater acoustic impulsive source. Shown are the (a) true and estimated first channel output, (b) first channel impulse
response, and (c) source. There are obvious magnitude and time shift differences, but the estimated
channel output and channel impulse response are remarkably similar in structure and potentially
useful for studying the underwater channel.

channel are mainly interested in the time
delays between peaks and relative phase
shifts [11]. Besides the overall magnitude ambiguity and time shift, it appears
we can accurately estimate the time
delays and phase shifts between the
peaks of the channel response.

Discussion
We have demonstrated a framework for
computing the adjoint of discrete wave140

let transforms, which arise in gradient
computations in optimization problems. The framework separates the
action of the wavelet transform and the
signal extension operator. The halfpoint symmetric extension is a common choice, and we have provided its
transpose, but other extension operators
can readily be cast into the framework.
For many practical purposes, it appears
that the adjoint operator is remarkably
IEEE Signal Processing Magazine

November 2016

James Folberth (james.folberth@
colorado.edu) is a Ph.D. student in
Stephen Becker's Information Extraction
group in the Department of Applied
Mathematics at the University of
Colorado at Boulder. His research interests include signal and image processing,
machine learning, and optimization.
Stephen Becker (stephen.becker@
colorado.edu) received his bachelor's degree in mathematics and physics from
Wesleyan University in 2005 and his
Ph.D. degree in 2011 in applied and computational mathematics from the
California Institute of Technology. He is
an assistant professor in the Department of
Applied Mathematics at the University of
Colorado at Boulder. He was the recipient
of a postdoctoral fellowship to study at
Paris 6 and the Goldstine postdoctoral fellowship at IBM Research. His work on
the TFOCS algorithm won the triennial
Beale-Orchard-Hays Prize from the
(continued on page 147)

http://https:// http://www.github.com/jamesfolberth/ieee_

Table of Contents for the Digital Edition of Signal Processing - November 2016