\begin{slide}
{}
\begin{center}
{\bf Overview}
\end{center}
{\tiny
\begin{itemize}
\item Representation of Signals
   \begin{itemize}
   \item Bode
   \item Phase and Group Delay
   \item Sampling
   \item Decimation and Interpolation
   \item FFT
   \item Convolution
   \item Chirp-Z Transform
   \end{itemize}
\item FIR Filter Design
   \begin{itemize}
   \item Windowing
   \item Optimal Minimax
   \end{itemize}
\item IIR Filter Design
   \begin{itemize}
   \item Analog
      \begin{itemize}
      \item Butterworth
      \item Chebyshev
      \item Elliptic
      \end{itemize}
   \item Discrete
   \end{itemize}
\item Spectral Estimation
   \begin{itemize}
   \item Modified Periodogram Method
   \item Correlation Method
   \item Maximum Entropy Method
   \end{itemize}
\item Kalman and Wiener Filtering
\end{itemize}
}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\begin{center}
{\bf Bode Plots}
\end{center}
\vfill
\vbox{
{\tiny
DEFINITION
%
$$
M(\omega)=20\log_{10}|H(s)_{s=j\omega}|
$$
%
\vfil
%
$$
\Theta(\omega)=\tan^{-1}[\frac{Im(H(s)_{s=j\omega})}{Re(H(s)_{s=j\omega})}]
$$
%
\vfil\vfil
LET
%
$$
H(s)=C\frac{\prod_{n=1}^{N}(s-a_n)}{\prod_{m=1}^{M}(s-b_m)}
$$
%
\vfil\vfil
THEN
%
$$
M(\omega)=\sum_{n=1}^{N}20\log\sqrt{\omega^2+a_n^2}-\sum_{m=1}^M\sqrt{\omega^2+b_m^2}
$$
%
\vfil
%
$$
\Theta(\omega)=\sum_{n=1}^{N}\tan^{-1}(\omega/(-a_n))-\sum_{m=1}^M\tan^{-1}(\omega/(-b_m))
$$
%
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/bode.1.ps}
\end{picture}
\center{\tiny Log-Magnitude Plot of $H(s)=1/(s-a)$}
}
\vfil
\center{
\begin{picture}(360,204)
\special{../figs/bode.2.ps}
\end{picture}
\center{\tiny Phase Plot of $H(s)=1/(s-a)$}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\begin{center}
{\bf Phase and Group Delay}
\end{center}
\vfill
\vbox{
{\tiny
DEFINITION
%
$$
H(\omega)=A(\omega)e^{j\theta(\omega)}
$$
%
\vfil
%
$$
\Theta(\omega)=\tan^{-1}[\frac{Im(H(s)_{s=j\omega})}{Re(H(s)_{s=j\omega})}]
$$
%
\vfil
%
$$
t_p(\omega)=\theta(\omega)/\omega
$$
%
\vfil
%
$$
t_g(\omega)=d\theta(\omega)/d\omega.
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/group.1.ps}
\end{picture}
\center{\tiny Modulated Exponential Signal}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/group.2.ps}
\end{picture}
\center{\tiny Constant Phase Band Pass Filter}
}
\vfil
\center{
\begin{picture}(360,204)
\special{../figs/group.3.ps}
\end{picture}
\center{\tiny Carrier Phase Shift by $t_p=\pi/2$}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/group.4.ps}
\end{picture}
\center{\tiny Linear Phase Band Pass Filter}
}
\vfil
\center{
\begin{picture}(360,204)
\special{../figs/group.5.ps}
\end{picture}
\center{\tiny Envelope Phase Shift by $t_g=-1$}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\begin{center}
{\bf Sampling}
\end{center}
\vfill
\vbox{
{\tiny
%
$$
x(n)=x(t)|_{t=nT}
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/sample.1.ps}
\end{picture}
\center{\tiny Frequency Response $X(\Omega)$}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/sample.2.ps}
\end{picture}
\center{\tiny Frequency Response $x(\omega)$ With No Aliasing}
}
\vfil
\center{
\begin{picture}(360,204)
\special{../figs/sample.3.ps}
\end{picture}
\center{\tiny Frequency Response $x(\omega)$ With Aliasing}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/sample.4.ps}
\end{picture}
\center{\tiny Cosine Signal}
}
\vfil
\center{
\begin{picture}(360,204)
\special{../figs/sample.5.ps}
\end{picture}
\center{\tiny Aliased Cosine Signal}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\begin{center}
{\bf Decimation and Interpolation}
\end{center}
\vfill
\center{
\begin{picture}(360,204)
\tenrm
\put(10,120){\makebox(0,0){x(nT)}}
\put(25,120){\vector(1,0){10}}
\put(35,100){\framebox(65,40){\shortstack{Put\\L-1 Zeros\\Between\\Each Sample}}}
\put(100,120){\vector(1,0){15}}

\put(115,100){\framebox(65,40){LPF}}
\put(180,120){\vector(1,0){15}}

\put(195,100){\framebox(65,40){\shortstack{Discard\\M-1 of Every\\M Samples}}}
\put(260,120){\vector(1,0){10}}
\put(300,120){\makebox(0,0){x(nMT/L)}}

\end{picture}
}
\center{\tiny Block Diagram of Interpolation and Decimation}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\begin{center}
{\bf DFT and FFT}
\end{center}
\vfill
\vbox{
{\tiny
DEFINITION
%
$$
X(k)=\sum_{n=0}^{N-1}x(n)e^{-j\frac{2\pi}{N}nk}
$$
%
\vfil
%
$$
x(n)=\frac{1}{N}\sum_{k=0}^{N-1}X(k)e^{j\frac{2\pi}{N}nk}
$$
%
\vfil
MATRIX FORMULATION
%
$$
X=Fx
$$
$$
F=\left[\begin{array}{ccccc}
1&1&1&\cdots&1\\
\\
1&e^{-j\frac{2\pi}{N}}&e^{-j\frac{4\pi}{N}}&\cdots&e^{-j\frac{2(N-1)\pi}{N}}\\
\\
1&e^{-j\frac{4\pi}{N}}&e^{-j\frac{8\pi}{N}}&\cdots&e^{-j\frac{4(N-1)\pi}{N}}\\
\\
\vdots&\vdots&\vdots&&\vdots\\
\\
1&e^{-j\frac{2(N-1)\pi}{N}}&e^{-j\frac{4(N-1)\pi}{N}}&\cdots&e^{-j\frac{(N-1)^2\pi}{N}}\\
\end{array}\right]
$$
%
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\begin{center}
{\bf Convolution}
\end{center}
\vfill
\vbox{
{\tiny
DEFINITION
%
$$
y(n)=\sum_{k=0}^{n}h(n-k)x(k)
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\tenrm
\put(65,120){\makebox(0,0){x(n)}}
\put(80,120){\vector(1,0){35}}

\put(115,100){\framebox(65,40){h(n)}}
\put(180,120){\vector(1,0){35}}

\put(270,120){\makebox(0,0){y(n)=h(n)*x(n)}}

\end{picture}
\center{\tiny Convolution Performed by Linear System}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Representation of Signals}}
\begin{center}
{\bf Chirp-Z Transform}
\end{center}
\vfill
\vbox{
{\tiny
DEFINITION
%
$$
\begin{array}{cc}
{\displaystyle X(z_k)=\sum_{n=0}^{N-1}x(n)A^{-n}W^{nk}}, & k=0,1,\ldots,M-1
\end{array}
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/czt.1.ps}
\end{picture}
\center{\tiny Samples of the z-transform on Spirals}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{FIR Filter Design}}
\begin{center}
{\bf Windowing}
\end{center}
\vfill
\vbox{
{\tiny
%
$$
H(\omega|\omega_c)=\left\{ \begin{array}{ll}
                     1, & \mbox{$|\omega|\leq\omega_c$}\\
                     0, & \mbox{otherwise}
                \end{array}
       \right.
$$
%
\vfil
%
$$
\begin{array}{cc}
h(n|\omega_c)=\frac{1}{\pi n}\sin(\omega_cn) &   -\infty<n<\infty
\end{array}
$$
%
\vfil
%
$$
R_N(n)=\left\{ \begin{array}{ll}
                     1, & \mbox{$0 \leq n \leq N-1$}\\
                     0, & \mbox{otherwise}
                \end{array}
       \right.
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/fir.1.ps}
\end{picture}
\center{\tiny Rectangularly windowed low-pass filter}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{FIR Filter Design}}
\begin{center}
{\bf Optimal Minimax}
\end{center}
\vfill
\vbox{
{\tiny
%
$$
E(f)=W(f)(D(f)-H(f))
$$
%
\vfil
%
$$
H^*(f)=\arg \min_{H(f)}\|E(f)\|_{\infty}
$$
%
\vfil
%
$$
H(f)=\sum_{n=-N}^{N}h_n e^{-j2\pi fn}
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/remez.3.ps}
\end{picture}
\center{\tiny Low Pass Filter with Transition Band $[.24,.26]$}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{IIR Filter Design}}
\begin{center}
{\bf Analog Butterworth}
\end{center}
\vfill
\vbox{
{\tiny
DEFINITION
%
$$
h_n^2(\omega \vert \omega_c)=\frac{1}{1+{(\frac{\omega}{ \omega_c})}^{2n}}
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/analog.1.ps}
\end{picture}
\center{\tiny Magnitude in dB. $n=13,\omega_c=300$}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{IIR Filter Design}}
\begin{center}
{\bf Analog Chebyshev Type I}
\end{center}
\vfill
\vbox{
{\tiny
DEFINITION
%
$$
 h_{1,n}^2(\omega \mid \omega_c , \epsilon)=\frac{1}{1+\epsilon^2 T_{n}^2(
\frac{\omega}{\omega_c})}
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/analog.3.ps}
\end{picture}
\center{\tiny Magnitude of a Type 1 Chebyshev filter}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{IIR Filter Design}}
\begin{center}
{\bf Analog Chebyshev Type II}
\end{center}
\vfill
\vbox{
{\tiny
DEFINITION
%
$$
 h_{2,n}^{2}(\omega \mid \omega_r , A ) = \frac{1}{1+\frac{A^2-1}{T_{n}^{2}(\frac{\omega_r}{\omega})}}
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/analog.5.ps}
\end{picture}
\center{\tiny Magnitude of a Type 2 Chebyshev filter}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{IIR Filter Design}}
\begin{center}
{\bf Analog Elliptic}
\end{center}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/analog.9.ps}
\end{picture}
\center{\tiny v(z) for z in $\Sigma_n$, with $n=9$}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{IIR Filter Design}}
\begin{center}
{\bf Discrete Filters by Bilinear Transformation}
\end{center}
\vfill
\vbox{
{\tiny
%
$$
H(s)=B(s)/A(s)
$$
%
\vfil
%
$$
s=\frac{1-z^{-1}}{T}
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/iir.1.ps hoffset=60}
\end{picture}
\center{\tiny Transform $s=(1-z^{-1})/T$}
}}
\vfill
\vbox{
{\tiny
%
$$
s=\frac{2}{T}[\frac{1-z^{-1}}{1+z^{-1}}]
$$
%
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Spectral Estimation}}
\begin{center}
{\bf The Problem}
\end{center}
\vfill
\vbox{
{\tiny
%
$$
S_x(\omega)=\frac{1}{N}|\sum_{n=0}^{N-1}x(n)e^{-j\omega n}|^2
$$
%
\vfil
%
$$
S_x(\omega)=\sum_{m=-\infty}^{\infty}R_x(m)e^{-j\omega m}
$$
%
\vfil
%
$$
\hat{R}_x(m)=\frac{1}{N}\sum_{n=0}^{N-1-m}x(n+m)x^*(n)
$$
%
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/spect.1.ps}
\end{picture}
\center{\tiny Overlapping Data}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Spectral Estimation}}
\begin{center}
{\bf Modified Periodogram Method}
\end{center}
\vfill
\vbox{
{\tiny
%
$$
I(\omega)=\frac{1}{U}|\sum_{n=0}^{N-1}w(n)x(n)e^{-j\omega n}|^2.
$$
%
\vfil
%
$$
\hat{S}_x(\omega)=\frac{1}{K}\sum_{k=0}^{K-1}I_k
$$
%
}}
\vfill
\begin{center}
{\bf Correlation Method}
\end{center}
\vfill
\vbox{
{\tiny
%
$$
\hat{R}_k(m)=\sum_{n=0}^{N-1-m}x(n+m)x^*(n)
$$
%
\vfil
%
$$
\hat{S}_x(\omega)={\cal F}\{\tilde{R}_x(m)w(m)\}
$$
%
\vfil
%
$$
\tilde{R}_x=\frac{1}{K}\sum_{k=1}^{K}\hat{R}_k
$$
%
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Spectral Estimation}}
\begin{center}
{\bf Maximum Entropy Method}
\end{center}
\vfill
\vbox{
{\tiny
%
$$
\hat{S}_x(\omega) = 
\max_{S(\omega)}\{-\int_{-\pi}^{\pi}S(\omega)\log[S(\omega)]d\omega\}
$$
%
\vfil
%
$$
\begin{array}{cc}
{\displaystyle 
\frac{1}{2\pi}\int_{-\pi}^{\pi}\hat{S}_x(\omega)e^{j\omega n}d\omega 
= \hat{r}_x(n)}, & \mbox{$n=0,1,\ldots,N-1$}
\end{array}
$$
%
\vfil
%
$$
\begin{array}{cc}
     \hat{S}_x(\omega)=
\frac{\sigma^2}{|1+\sum_{n=1}^{N-1}a_n\exp\{-j\omega n\}|^2}
\end{array}
$$
%
\vfil
%
$$
\begin{array}{cc}
{\displaystyle \hat{r}_x(n) = -\sum_{k=1}^{N-1}a_k\hat{r}_{n-k}}, & n\ge N
\end{array}
$$
%
\vfil
%
$$
\begin{array}{cc}
\left[ \begin{array}{cccc}
\hat{r}_x(0) & \hat{r}_x(1) & \cdots & \hat{r}_x(N-1) \\
\hat{r}_x(1) & \hat{r}_x(0) & \cdots & \hat{r}_x(N-2) \\
\vdots       & \vdots       &        & \vdots \\
\hat{r}_x(N-1) & \hat{r}_x(N-2) & \cdots & \hat{r}_x(0) 
\end{array}\right]
\left[ \begin{array}{c}
1\\
a_1\\
\vdots\\
a_{N-1}
\end{array}\right]
=
\left[ \begin{array}{c}
\sigma^2\\
0\\
\vdots\\
0
\end{array}\right]
\end{array}
$$
%
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Spectral Estimation}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/mem.1.ps}
\end{picture}
\center{\tiny Input Data Sequence, $x(n)$}
}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/mem.2.ps}
\end{picture}
\center{\tiny Maximum Entropy Spectral Estimate of $x(n)$}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Spectral Estimation}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/mem.3.ps}
\end{picture}
\center{\tiny Squared Magnitude of the Fourier Transform of $x(n)$}
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Kalman and Wiener Filtering}}
\begin{center}
{\bf Kalman Filter}
\end{center}
\vfill
\vbox{
{\tiny
%
\begin{eqnarray}
\hat{x}_{k+1|k}&=&F_k\hat{x}_{k|k-1}+F_kK_k(y_k-H_k\hat{x}_{k|k-1})\nonumber\\
P_{k+1|k}&=&F_kP_{k|k-1}F_k^T-F_kK_kH_kP_{k|k-1}F_k^T+G_kQ_kG_k^T.
\end{eqnarray}
%
\vfil
%
\begin{eqnarray}
\hat{x}_{0|-1}&=&m_0\nonumber\\
P_{0|-1}&=&\Pi_0
\end{eqnarray}
%
\vfil
%
$$
K_k=P_{k|k-1}H_k^T[H_kP_{k|k-1}H_k^T+R_k]^{-1}
$$
%
\vfil
%
$$
\hat{x}_{k|k}=(I-KH)F\hat{x}_{k-1|k-1}+Ky_k
$$
%
}}
\vfill
\begin{center}
{\bf Wiener Filter}
\end{center}
\vfill
\vbox{
{\tiny
%
\begin{eqnarray}
\hat{x}(t)=\int_{a}^{b}H(t,s)y(s)ds
\end{eqnarray}
%
\vfil
%
$$
0=E\{(x(t)-\hat{x}(t))y^T(u)\}=R_{xy}(t,u)-\int_{a}^{b}H(t,s)R_{yy}(s,u)ds
$$
%
}}
\vfill
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
{}
{\tiny\underline{Kalman and Wiener Filtering}}
\vfill
\vbox{
\center{
\begin{picture}(360,204)
\special{../figs/wf.1.ps}
\end{picture}
\center{\tiny Wiener Smoothing Filter}
}}
\vfill
\end{slide}
