Background

The background material required to go through this course is some undergraduate level understanding of signals and systems, graduate level understanding probability and random processes and basic linear algebra and vector space concepts that we typically cover in ECEN 601. We will try to summarize the main results that are required in these areas.

Signals and Systems

We will encounter several kinds of signals such as transmitted analog waveforms, received analog waveforms, discrete-time data sequences etc. So it is best to start with a quick review of basic results from signals and systems. We will talk about integrals of signals, Fourier transforms of signals, correlation (inner product) between two signals, energy of a signal etc without worrying about whether these integrals, Fourier transforms or inner products are well defined. A mathematically minded student who is unhappy with this approach can find a justification here.


Description of commonly used signals:

Knowing the definitions of commonly used signals such as the unit step function, rectangular function, dirac-delta function and the sinc function will be useful.


Fourier Transform and Commonly used FT Pairs

We will consider the Fourier transform where frequency is in Hertz. For a signal $s(t)$, its Fourier transform will be denoted by $S(f)$ and they are related according to

(2)
\begin{eqnarray} \nonumber S(f) & = & \int_{-\infty}^{\infty} s(t) e^{-j 2 \pi f t} \ dt \\ \nonumber s(t) & = & \int_{-\infty}^{\infty} S(f) e^{j 2 \pi f t} \ dt \end{eqnarray}

Properties of the Fourier Transform

The following properties of the Fourier transform will be useful.


Energy and Power of Signals

Th energy $E_s$ and power $P_s$ of a signal $s(t)$ are defined as

(12)
\begin{eqnarray} \nonumber E_s & = & \int |s(t)|^2 \ dt \\ \nonumber P_s & = & \lim_{T \rightarrow \infty} \frac{1}{T} \int_{-T/2}^{T/2} |x(t)|^2 \ dt \end{eqnarray}

A signal is said to be an energy type signal if $E_s$ is bounded and it is said to be a power type signal if the power is non-zero and bounded.


Correlation Function and Matched Filter

For any two deterministic signals Energy-type signals $s_1(t)$ and $s_2(t)$, the crosscorrelation function between them is given by

(13)
\begin{align} R_{s_1,s_2}(\tau) = \int s_1(u) \ s_2^*(t-u) \ du \end{align}

Notice that the above correlation function can be computed by passing $s_1(t)$ through a linear time invariant filter with impulse response $h(t) = s_2^*(-t)$ as shown in the figure. Such a filter is called a filter that is matched to the signal $s_2(t)$. This view point is very useful in this course and we will have more to say about this later.

correlationmatchedfilter.png

For the special case when $s_2(t) = s_1(t)$, the crosscorrelation function is called the autocorrelation function and is denoted as $R_{ss}(\tau)$.

For power-type signals the correlation function needs to be defined as

(15)
\begin{align} R_{s_1,s_2}(\tau) = \lim_{T \rightarrow \infty} \frac{1}{T} \int_{-T/2}^{T/2} s_1(t) \ s_2^*(t-\tau) \ dt \end{align}

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