2.2 lecture 2, Wednesday January 27, 2010

Define energy of signal as \(E= \sum \limits _{n=-\infty }^{\infty } \left \vert x(n) \right \vert ^{2}\)

Showed that any signal \(x\left ( n\right ) \) can be written as sum of weighted shifted Dirac delta functions, hence \[ x\left ( n\right ) ={\sum \limits _{k=-\infty }^{\infty }}x\left ( k\right ) \delta \left ( n-k\right ) \]

Using the above, derived the convolution equation for linear time invariant system as follows. Let system be \(T\), hence we have

\begin{align*} y\left ( n\right ) & =T\left [ x\left ( n\right ) \right ] \\ & =T\left [{\sum \limits _{k=-\infty }^{\infty }}x\left ( k\right ) \delta \left ( n-k\right ) \right ] \\ & ={\sum \limits _{k=-\infty }^{\infty }}T\left [ x\left ( k\right ) \delta \left ( n-k\right ) \right ] \end{align*}

Now, for linear system, we know that \(T\left [ af\left ( n\right ) \right ] =aT\left [ f\left ( n\right ) \right ] \) for constant \(a\), hence the above becomes

\[ y\left ( n\right ) ={\sum \limits _{k=-\infty }^{\infty }}x\left ( k\right ) T\left [ \delta \left ( n-k\right ) \right ] \]

Now, let the response of the system for \(\delta \left ( n-k\right ) \) be called \(h\left ( n,k\right ) \), hence the above becomes

\[ y\left ( n\right ) ={\sum \limits _{k=-\infty }^{\infty }}x\left ( k\right ) h\left ( n,k\right ) \]

Now, assuming this is an LTI system, then the time when the impulse occurred would not change the response, hence the above becomes

\[ y\left ( n\right ) ={\sum \limits _{k=-\infty }^{\infty }}x\left ( k\right ) h\left ( n-k\right ) \]

And the above is the convolution equation, written as \(y\left ( n\right ) =x\left ( n\right ) \circledast h\left ( n\right ) \)

Next, went over definition of linear system. Linear system is one where if \(x_{1}\rightarrow y_{1}\) and \(x_{2}\rightarrow y_{2}\) then output of system when the input is \(ax_{1}+bx_{2}\) must be the same as \(ay_{1}+by_{2}\). If it is not the same, then the system is not linear. Here, I assumed \(a,b\) are constants, and I meant by \(x_{i}\) as the input and by \(y_{i}\) as the output.

Next, went over how to check if system is linear or not. see my study notes. Next went over definition of time invariant, which is: Output of a delayed input is the same as delayed output of the input. (delay amount is same ofcourse). Then went over how to check for time invariant. see my study notes.

Final went over linear convolution of 2 sequences and showed an example. Easy to do. Flip \(h\left ( n\right ) \), the move the flipped \(h\left ( h\right ) \) sequence and slide it over \(x\left ( n\right ) \). Each time multiply corresponding values and adding.

That was the end of the second lecture.