Minor errata cleared up

A variety of minor errata have been taken care of. Still have some
medium-sized improvements to make for fall 2016. And then there is the
bigger stuff…

Author: stiber

Signal Computing.tex

@@ -210,7 +210,7 @@
 {\normalsize Prof. Eric C. Larson}\\
 {\normalsize Southern Methodist University}}
 
-% \date{August 2015}
+% \date{August 2016}
 
 % \includeonly{ch-fft/fft} % ch-iir/fb-filters} % ch-conv/convolution}
 % %
@@ -229,7 +229,7 @@
 
 \mbox{}\hfill
 \parbox{1.5in}{\mbox{}}
-\parbox{3in}{Fall 2015}\\[0.25in]
+\parbox{3in}{Fall 2016}\\[0.25in]
 \mbox{}\hfill
 \parbox{3in}{\raggedright
 Michael Stiber\\
@@ -247,7 +247,7 @@
 \newpage
 \mbox{}\vspace{2in}
 \begin{flushright}
-Copyright \copyright\ 2002--2015 by Michael and Bilin Stiber and Eric
+Copyright \copyright\ 2002--2016 by Michael and Bilin Stiber and Eric
 C. Larson\\[1in]
 This material is based upon work supported by the National Science
 Foundation under Grant No. 0443118.\\[2in]

ch-av/audio-video.tex

@@ -361,7 +361,7 @@ \subsection{JPEG}
 the basic MATLAB distribution has no 2D DCT built in. The resulting
 complex output was converted to reals using the \verb|abs()| function
 (the MATLAB code for all this is located at
-\url{http://courses.washington.edu/css457/ebook/dctdemo.m}. The top
+\url{http://faculty.washington.edu/stiber/pubs/Signal-Computing/}. The top
 left image is a pixel block in which the pixel values vary in
 intensity as the sine of the x coordinate only.  We would expect then
 that it would have nonzero spectral components for $k>0$, because

ch-computer/computer-signals.tex

@@ -108,7 +108,7 @@ \section{From the physical to the digital}
 --- converting an analog electrical signal into a sequence of binary
 numbers.  Digitization involves two processes: \emph{sampling} and
 \emph{quantization}.  In the former process, the value of the analog
-signal is measures at regular intervals of time (the \emph{sampling
+signal is measured at regular intervals of time (the \emph{sampling
 interval}). The output of a sample and hold (S/H) device will
 maintain a fixed level in between sampling times. This makes the
 quantization process easier: the analog-to-digital converter (ADC)
@@ -147,9 +147,9 @@ \section{Measuring Noise}
 \index{signal-to-noise ratio (SNR)}
 \index{decibel}
 Mathematically, SNR becomes:
-\[
+\begin{equation}
 \text{SNR}=20\log_{10}\left(\frac{x_\mathrm{RMS}}{n_\mathrm{RMS}}\right) \label{eq:SNR}
-\]
+\end{equation}
 We can quantify the amount of noise in a signal by using the SNR. The
 idea is that noise becomes more of a problem when it reaches the same
 magnitude as the signal. For example, a slight hissing noise on a
@@ -201,7 +201,7 @@ \section{Sampling}
 \index{analog-to-digital conversion!aliasing|(}
 We've already stated that we'd like the digitization process to retain
 all the information in the original physical signal (or, at least,
-that not being possible, to retain all information can be practically obtained).
+that not being possible, to retain all information as far as is practical).
 However, sampling alone can destroy information. In the case of
 figure~\ref{fg:sampled} (top), no information is lost (the original
 waveform could in principle be reconstructed) because the sampling
@@ -237,7 +237,7 @@ \subsection{Aliasing}
 sample times: the phasor represented as a sequence of measurements,
 each taken at a time that is a multiple of the sampling interval
 $T_s$. This was done by replacing $t$ by $nT_s$, the sampling times
-($\{0, T_s, 2T_s, 3T_s, \ldots$). $x[n]$ is a \emph{discrete} signal
+$\{0, T_s, 2T_s, 3T_s, \ldots\}$. $x[n]$ is a \emph{discrete} signal
 (a function of discrete time). For example, $x[n]$ could be the black circles in
 figure~\ref{fg:sampled}. The value of $x[n]$ is undefined when $n$ is not an integer. 
 

ch-conv/convolution.tex

@@ -416,8 +416,8 @@ \subsection{Example: z-transform of exponential signal}
 \label{eq:zt-expo-cF}
 \end{equation}
 
-Obviously $z=Re^{j\theta}$ is the pole (a zero in the denominator;
-you'll learn more about this in
+When $z=Re^{j\theta}$ we have what we call a \emph{pole} (a zero in the
+denominator; you'll learn more about this in
 Chapter~\ref{ch:fb-filters}). Equation~(\ref{eq:zt-expo-cF}) can be
 broken into real and imaginary parts using Euler's formula:
 \begin{align}
@@ -514,10 +514,10 @@ \section{Convolution}
 \item Associative : $(X \ast H) \ast G = X \ast (H \ast G)$
 \index{convolution!associative property}
 \end{enumerate}
-Obviously, $x \ast \vec{0} = \vec{0} \ast x = 0$ (where $\vec{0}$ is a
-vector of all zeros). Then how about $\vec{1} \ast \vec{1}$ (where
-$\vec{1}$ is a vector of all ones, in this case $\vec{1}=u[n]$; see
-the self-test exercise)?
+You should be able to convince yourself that $x \ast \vec{0} = \vec{0}
+\ast x = 0$ (where $\vec{0}$ is a vector of all zeros). How about
+$\vec{1} \ast \vec{1}$ (where $\vec{1}$ is a vector of all ones, in
+this case $\vec{1}=u[n]$; see the self-test exercise)?
 
 \subsection{Example of Convolution}
 

ch-fft/fft.tex

@@ -185,9 +185,9 @@ \section{The Discrete Fourier Transform}
 \label{eq:idft}
 \end{equation}
 
-Obviously, the DFT converts a finite (periodic), discrete signal in
-the time domain into a finite, discrete spectrum in the frequency
-domain.
+The DFT converts a finite (periodic), discrete signal in
+the time domain ($x[n]$) into a finite, discrete spectrum in the frequency
+domain ($X[k]$).
 
 We can derive the DFT informally from the Fourier series of a periodic
 signal. Let's compare equation~(\ref{eq:dft}) and that for the
@@ -381,7 +381,7 @@ \subsection{Properties of the DFT}
     \overbrace{2 \quad \underbrace{3 \quad 4 \quad 5} \quad 6} \quad 7}
 \label{eq:dft8-freqs}
 \end{equation}
-Obviously, it is symmetric about $k=4$. When $N$ is an even number,
+This is clearly symmetric about $k=4$. When $N$ is an even number,
 the spectrum has $N/2$ at the center; when $N$ is an odd number, the
 center is between $\lfloor N/2 \rfloor$ and $\lceil N/2 \rceil$. So,
 for $N=7$,
@@ -1283,7 +1283,7 @@ \subsubsection{Hamming Window}
           \quad n=0,1,\ldots,L-1
 \label{eq:ufft-hnw}
 \end{equation}
-It consists of a cycle of a cosine, droping to 0.08 at the end-points
+It consists of a cycle of a cosine, dropping to 0.08 at the end-points
 and with a peak value of one. A plot of a Hamming window is shown in
 figure~\ref{fig:ufft-hmw}
 

ch-fir/filt-intro.tex

@@ -138,7 +138,7 @@ \subsection{Delaying a phasor}
 \emph{phasor}, a complex sinusoid expressed as $e^{j\omega t}$.  We
 also saw that this representation makes the math simpler for adding
 sinusoids, at least.  A phasor's magnitude is one, its frequency is
-$\omega$, and it angle is $\omega t$, where $t$ is time. It moves
+$\omega$, and its angle is $\omega t$, where $t$ is time. It moves
 around the unit circle counter-clockwise along time.  If we delay it
 by $\tau$ sec, then the delayed time is $t-\tau$, and we can write
 this as:
@@ -609,7 +609,7 @@ \subsubsection{Self-Test Exercises}
 See~\ref{sc:ch3ex} \#\ref{it:ch3ex3}--\ref{it:ch3ex5} for answers.
 
 \begin{enumerate}
-\item Write equation~(\ref{eq:ff-manyk}) for $k=0, 1, 2, 3$, then
+\item Write equation~(\ref{eq:ff-manyk}) for $M=0, 1, 2, 3$, then
   write the transfer function for each.
 \item Given the signal $x(t) = \sin t$ and the derivative operator
   $D=\derivin{}{t}$, what is $Dx(t)$?
@@ -650,7 +650,7 @@ \subsection{The z-plane}
 \begin{equation}
 Y = [1-b_1z^{-1}]X = H(z)X
 \end{equation}
-Obviously,
+We can express the transfer function as
 \begin{equation}
 H(z) = 1-b_1z^{-1} = (1-b_1z^{-1}) \frac{z}{z} = \frac{z-b_1}{z}
 \end{equation}
@@ -688,7 +688,7 @@ \subsection{The z-plane}
 minimum.
 \index{zero}
 \index{frequency response!zero}
-Obviously, when the zero ($b_1$) is near $\hat{\omega}=0$ $(z=1)$,
+When the zero ($b_1$) is near $\hat{\omega}=0$ $(z=1)$,
 this results in a high pass filter because it doesn't pass frequencies
 near zero (low frequencies). Similarly, when the zero ($b_1$) is near
 $\hat{\omega}=\pi$ $(z=-1)$ we obtain a low pass filter: high frequency
@@ -797,7 +797,7 @@ \subsection{Phase Response}
 \index{filter!phase response|(emph}
 So far we have been talking about the magnitude response of
 $\mathcal{H}(\hat{\omega})$.  In the last topic of this chapter, let's
-talk about is its phase response. We already know that
+talk about its phase response. We already know that
 $\mathcal{H}(\hat{\omega})$ can be expressed in polar form, with its
 magnitude and angle
 \begin{equation}
@@ -889,9 +889,10 @@ \subsection{Phase Response}
 delay it by one sampling interval (from the $(n-1)$ term in the
 exponential) and to multiply it by $2\cos\hat{\omega}$. Notice that
 the delay is independent of its frequency. When all the frequency
-components of a signal are delayed by an equal amount, say the filter
-has \emph{no phase distortion} or \emph{linear phase}. The magnitude response and phase response are
-shown in figure~\ref{fig:ff-exp2zh90-r1}.
+components of a signal are delayed by an equal amount, we say the
+filter has \emph{no phase distortion} or \emph{linear phase}. The
+magnitude response and phase response are shown in
+figure~\ref{fig:ff-exp2zh90-r1}.
 
 \begin{figure}
 \centerline{\includegraphics[width=6in]{ch-fir/ffexp_2tdelay_h90}}

ch-iir/fb-filters.tex

@@ -71,7 +71,8 @@ \subsection{Poles}
   y[n] = b_0 x[n] + b_1 x[n-1] + a_1 y[n-1]
   \label{eq:fb-ff}
 \end{equation}
-but for the moment we will concentrate on just the feedback.
+but for the moment we will concentrate on just the feedback part of
+the filter, from equation~(\ref{eq:fb-1p}).
 
 From equation~(\ref{eq:fb-1p}), the z-transforms of the input and
 output signals, and the delay operator $z^{-1}$, we can get the
@@ -289,7 +290,8 @@ \subsection{Stability}
 \begin{equation}
 1, p_i, p_i^2, p_i^3, \ldots
 \end{equation}
-Obviously, the impulse response is stable only when the pole's $|p_i|<1$
+This impulse response is stable only when the pole's $|p_i|<1$, so
+that the sequence is decreasing in magnitude.
 
 \item \textbf{An impulse response is neither stable nor unstable if
 $|p|=1$.}

ch-physical/physical-signals.tex

@@ -1089,7 +1089,7 @@ \subsection{Derivation of the Fourier Series}
 = 1 \vec{\mathbf{x}} + 2 \vec{\mathbf{y}} + 3 \vec{\mathbf{z}}$).
 This is the signal's representation in the \emph{frequency domain},
 where frequency is the coordinate (in other words, the signal
-expressed as a function of frequency).  Similar, $f(t)$ is the
+expressed as a function of frequency).  Similarly, $f(t)$ is the
 signal's \emph{time domain} representation, where time is the
 coordinate. Notice that the Fourier series transforms a finite
 (periodic), continuous signal in the time domain into an infinite

intro.tex

@@ -182,8 +182,8 @@ \section*{About This Book}
 \url{http://faculty.washington.edu/stiber/pubs/Signal-Computing/},
 including a pointer to its source material on GitHub. If you find any
 errors, make any changes, or find this book useful in your course or
-your life, please email us at \url{mailto:[email protected]}, or send us a
-pull request.
+your life, please email us at
+\href{mailto:[email protected]}{[email protected]}, or send us a pull request.
 
 \subsection*{Typographical Conventions}
 \addcontentsline{toc}{section}{Typographical Conventions}
@@ -204,6 +204,8 @@ \subsection*{Typographical Conventions}
   $\omega$ & continuous angular frequency; units radians/second \\
   $\omega_0$ & a particular angular frequency; subscript may vary
                depending on use \\
+  $\omega'$ & apparent (continuous angular) frequency, as a result of
+                        aliasing \\
   $n$    & discrete time or sample number; dimensionless, but you can
            think of the units as being ``samples'' \\
   $x[n]$ & a function of discrete time; may be real-valued (sampled