Blockish Examples (note)

Math Examples

These are examples of math in the text.

MathML with ASCIIMath in it: $sq(t)\; sum\_(n=-infty)^infty\; (-1)^n\; A\; p\_\{p/2\}\{t-n\{T/2\}\}{\displaystyle sq\left(t\right)\sum _{n=-\infty }^{\infty }{(-1)}^{n}A{p}_{\frac{p}{2}}\{t-n\left\{\frac{T}{2}\right\}\}}$

MathML with TeX in it: $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$

MathML with Content Math inside it: $x^2$

Linear and Time-Invariant Systems

Signals

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A signal's complexity is not related to how wiggly it is. Rather, a signal expert looks for ways of decomposing a given signal into a sum of simpler signals, which we term the signal decomposition. Though we will never compute a signal's complexity, it essentially equals the number of terms in its decomposition. In writing a signal as a sum of component signals, we can change the component signal's gain by multiplying it by a constant and by delaying it.

A Title for a Figure

More complicated decompositions could contain derivatives or integrals of simple signals. In short, signal decomposition amounts to thinking of the signal as the output of a linear system having simple signals as its inputs. We would build such a system, but envisioning the signal's components helps understand the signal's structure. Furthermore, you can readily compute a linear system's output to an input decomposed as a superposition of simple signals. `x+sqrt(1-x^2)`.

Here is some TEX math $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$.

Example

As an example of signal complexity, we can express the pulse `p_\Delta(t)` as a sum of delayed unit steps.

`p_\Delta(t)=u(t)-u(t-\Delta)`

Thus, the pulse is a more complex signal than the step. Be that as it may, the pulse is very useful to us.

Exercise

Express a square wave having period `T` and amplitude `A` as a superposition of delayed and amplitude-scaled pulses.

Solution

`sq(t) sum_(n=-infty)^infty (-1)^n A p_{p/2}{t-n{T/2}}`

That's it so far!