- Given the forward DFT, derive the IDFT.
- Show that the DFT and IDFT are periodic. What is the period?
- Note that H(n) are complex, and h(k) are real. What restrictions does this put on H(n)? Hint: The answer is: $H(n) = H^*(-n)$.

- The dot or inner product of a two vectors, $\mathbf a$ and $\mathbf b$ is the product of the lengths of the two vectors multplied by the cosine of the angle between them. $$\mathbf a \cdot \mathbf b = |\mathbf a||\mathbf b| cos \theta = \sum_{k=0}^{N-1} a_k b_k^*$$ If one of the vectors is a unit vector, then the inner product is the projection of the other vector in the direction of the unit vector. This means the inner product is a way of computing how much one vector is in the direction of another. If the vectors come from a A/D converter length of a vector is a measure of how loud or how much power there is in that signal that was sampled. The inner product tells us how much one vector is like another, or how much or one vector is in the other, how close they are to sounding alike. How could you interpret $$H(n) = \sum_{k=0}^{N-1} h(k) e^{-j2\pi nk/N}$$ as you think of this as an inner product?
- Suppose you make your sampling interval smaller and smaller, so that you are taking an infinite number of samples in an interval $T_0$. Suppose you were told that you could write a periodic function that repeats over $T_0$ with this sum, how would you find the coefficients $c_n$? $$ x(t) = x(t+T_0) = \sum_{n=-\infty}^{\infty} c_n e^{j2\pi nt/T_0}$$

- How would you interpret the $c_n$?
- If $x(t) \in \mathbb{R}$, what can you say about $c_n$? Hint: the answer is $c_n = c_{-n}^*$.
- If $x(t) = x(-t) = x(t+T_0)$, what can you say about $c_n$? What is $x(t) = -x(-t)$?
- Find the Complex Fourier Series for $sgn(cos(t))$.
- Plot $sgn(cos(t))$, the first few of the terms in its Fourier series and the sum of those terms.

- What is the significance of negative $n$ in the Complex Fourier Series? Why are negative $n's$ needed?
- What frequency is the cosine associated with the $n = -1$ for an even periodic function, $x(t) = x(-t) = x(t + T_0)$?
- Suppose you were going to approximate the formula for the coefficients of the Complex Fourier Series with a rectangular integration sum. What would you get at frequencies $k/T_0$, for $k \in {0, 1, 2, ..., (N-1)/2}$?
- How does this relate to the DFT? If you only needed a finite number of them, could you save yourself some effort calculating some Fourier series coefficients using this? Given the IDFT $$h(k) = 1/N \sum_{n=0}^{N-1} H(n) e^{j2\pi nk/N}$$ Can you say how is the IDFT related to $x(t)=x(t+T_0)$ , and its Fourier series coefficients, $c_n$ ?

Today we tackle the problem that Fourier series only work for periodic signals. It would be very nice if we could think of frequency components of any signal, periodic or not. What if we just let the period, $T_0$ get really large? For example, for a long time people thought the earth was flat, not round, and this was primarily because the circumfrence of the earth is so large.

- Write $x(t)=x(t+T_0)$ as the Fourier series and then substitute the formula you found for $c_n$ into that, being careful not to get dummy variables mixed up with real ones. It is better to use limits of $−T_0/2$ and $T_0/2$ rather the $0$ to $T_0$. Remember that $n$ ranges vastly. What do the following things approach in the limit as $T_0$ approaches $\infty$?$$ n/T_0$$ $$\sum_{n=-\infty}^{\infty} 1/T_0$$ $$T_0$$ $$\int_{-T_0/2}^{T_0/2} $$

Today we try to understand the $\delta (u)$, the "Impulse Function," also known as the "Dirac Delta Function. You can think of an impulse function as the limit of a sequence of functions that has the sifting property, $$f(x) =\int_{-\infty}^{\infty} f(u) \delta(u-x) du$$

- Find an impulse function identity by interchanging the order of integration of $x(t) = \mathscr F^{-1} [\mathscr F[x(t)]]$. This involves interchanging the order of limiting operations, so it may not work for all $x(t)$. When I do iffy things like this, I put a flag in my notes, and when I run into some issues later, I check to see if they could have been caused by my flagged operation.
- What does this say about the orthogonality of $e^{j2\pi f t}$?
- Show that the impulse is an even function, $\delta(u) = \delta(-u)$.
- Evaluate: $$ \int_{-\infty}^{\infty} \delta(u) du$$
- Note that there are some problems finding the "length" of the impulse function.
- Show $\delta (au) = 1/|a| \delta(u)$.

In this section we will investigate linear time invariant systems. The idea is to take a system, and output due to a set of basis functions and figure out what the output due to any input is. We use the idea that we can make the input up as a linear combination of the basis functions. We also use time invariance to determine that the whole set of basis functions is.

- Fill in the following table: |Input to the LTI System| Output | Reason | |---|---|---| |$\delta(t)$ | $h(t)$ | Given | |$\delta(t-t_0)$| | | | $x(t)$ | | | |$e^{j2\pi ft} $| | | |$X(f) e^{j2\pi f t}$| | | |$\int_{-\infty}^{\infty}X(f) e^{j2\pi f t} df$| | |

- What is the difference between the last line and the $x(t)$ line?

Today we will review and work on notebooks to try and be ready for submissions Saturday night at 6:00 p.m.

- Together, lets do the same RC circuit from class with an excitation of $e^{j2\pi f t}$.
- Use that to find the output due to a periodic wave, $x(t) = x(t+T_0)$.
- write a python script to plot the output from a square wave. Does it look right?

Each of the following are common operations we do to signals in a system. It is useful to know how doing something in time affects the frequency components, and vice versa.

Show that a time shift of $t_0$ multiplies frequency components by a phase factor of $e^{-j2\pi ft}$. For the Fourier series, the frequencies are discrete, given by $f = n/T_0$.

Show that you can shift a signal, $x(t)$, in frequency by an amount $f_0$, by multiplying it by $e^{j2\pi f_0 t}$. xWhat happens if you multiply $x(t)$ by a cosine wave at frequency $1/T_0$? How about a square wave? This is a really handy thing. We used this in the VNA last year.

Show that differentiation of $x(t)$ is the same as multiplying the frequency componets by $j2\pi f$.

What corresponds to filtering using a filter transfer function, $H(f)$?

What are the $H(f)$ and $h(t)$ for the the time delay and differentiation? Is there a transfer function and impulse response for frequency shifting?

What do the above properties mean for Fourier series, and for digital signal processing?

In [ ]:

```
```