0 Vectors & Signals

Babak Ayazifar

ayazifar@berkeley.edu

eecs66.org

ee66@berkeley.edu <- staff contact

take notes

Nomenclature

we are going to discuss a lot of amazing topics, and there is a necessary baselines to know to build upon. We have to learn the basics of the language we are using to communicate.

once we build the foundational mathematics, we will get to the exciting stuff

Vectors & Signals

you have seen cartesian vectors in linalg

\(\underline{X} = [x_1, \ldots, x_n] \in \mathbb{R}^n\) iff \(x_k \in \mathbb{R}\) where \(k = 1 \ldots n\)

\(\underline{X} = [x_1, \ldots, x_n] \in \mathbb{C}^n\) iff \(x_k \in \mathbb{C}\) where \(k = 1 \ldots n\)

\(\langle n \rangle\) means a set of \(n\) contiguous integers

in print form vectors will be in bold

Signals are Functions

signals are functions, we will use them interchangeably

\(x: A \to B\) where \(x\) is the name, \(A\) is the domain, and \(B\) is the codomain (range)

codomain is a superset of the range of values that \(x\) can take

if \(x: \mathbb{R} \to \mathbb{R}\)

for all \(t\) in the domain \(A\) \(x(t) = \cos(2\pi t)\)

if i wanted to be really crazy

\(x: \mathbb{R} \to [-1, 1]\)

then \(\mathbb{R}\) is the domain, and \([-1, 1]\) is the range

if i have set \(A\) and set \(B\) and i have element \(t\) in \(A\) then i can use the function to map \(t\) to \(f(t)\) in \(B\)

it's possible another point in the domain can map to the same one in the codomain, but we cannot have the opposite because it is against the definition of a function

if each one has a unique value in each set, domain to codomain, then the function is invertible, and we can use the footprint to create an inverse function

Signals

signals are functions, we have different kinds of signals we deal with

Continuous-time signal (CT) Signals
If it maps from \(\mathbb{R}\) to either \(\mathbb{R}\) or \(\mathbb{C}\)
\(x: \mathbb{R} \to \mathbb{R}\) or \(\mathbb{C}\)
- if it maps to \(\mathbb{R}\) it's "real valued"
- if it maps to \(\mathbb{C}\) it's "complex valued"
we won't use the term analogue very often in this class, just CT
\([\mathbb{R} \to \mathbb{R}]\): set of all real valued signals
\([\mathbb{R} \to \mathbb{C}]\): set of all complex valued signals
- it's important we become comfortable with the complex number system
Discrete time signals (DT) Signals
\(x: \mathbb{Z} \to \mathbb{R}\) or \(\mathbb{C}\)
- takes an integer and maps to real or complex values
- the domain is the set of all integers
we will use the notation \(x[n]\): value of the signal \(x\) at time (sample) integer \(n\)
- \(x[t]\) is the value of the signal at time \(t\)
- \(x\) is just the signal in entirety
an example would be
- for all \(n \in \mathbb{Z}\), \(x[n] = \cos(\frac{\pi}{4} n)\)
the way we plot these is different from a continuous variable
- we plot the \(n\) axis and then specify its value with lollipops

      x[n] = cos((pi/4) n),  n = 0..8

       1.0   o                               o
             |                               |
       0.7   |   o                       o   |
             |   |                       |   |
       0.0 --+---+---o---+---+---+---o---+---+--> n
                         |   |   |
      -0.7               o   |   o
                             |
      -1.0                   o
             0   1   2   3   4   5   6   7   8

- formally called a stem plot
- more often than not called lolli pop plot

\(x[0.5]\) is undefined because it maps only integers
- gotta be careful about the domain and codomain is, etc.
important discrete time signals
- we have a discrete time impulse (DT) impulse (Kronecker Delta)
- \(\delta[n]\)
- it's very simple, we use the value 1 at \(n = 0\) and is 0 at any other integer
- \(\delta[n] = 0\) at all other values
- this looks really innocent but it's really critical
- DT Unit Step
- as the name suggests this is a signal that undergoes a step jump at zero
- \(u[n] = 0\) at \(n < 0\)
- \(u[n] = 1\) at \(n \geq 0\)
operation of signals
- shift of signals
- lets use our test, the kronecker delta
- if \(\hat{\delta}[n] = \delta[n]\) is it shifted to the right or the left?
- shifted right, this is a unit delay, where it is one to the right
- if \(\tilde{\delta}[n] = \delta[n+1]\) is a unit advance.
- lets say \(x\) is a discrete time, real valued signal?
- what is \(x[n-0.5]\) this is always undefined because the shifts have to be integers
- Express the unit step as a linear combo of shifted impulses
- \(u[n] = \sum_{i=0}^{\infty} \lambda(n - i)\)
- this is one way we can express the unit step as a linear combo of impulses
- but we can generalize this now!
- bold claim coming in really quick.
- any DT signal can be expressed as a linear combination of shifted impulses
- \(x[n] = 3\delta[n+1] + \delta[n] - \delta[n-1] + 2\delta[n-2]\)
  - the coefficients are the value of the signal at each one.
  - we can write a more general form here
  - \(x[n] = \sum_{k=-\infty}^{\infty} x[k]\lambda[n-k]\)
- where do these dt signals come from?
- we are taking samples of analogue signals
- maybe other stuff too?
- closing price of a market
- changes of variables is represented as
- \(u[n] = \sum_{k=0}^{\infty} \lambda(n - k)\)
- if i let \(l\) be \(n-k\)
- then \(u[n] = \sum_{l=-\infty}^{n} \lambda(n - k)\)
  - cumulative sum is useful for calculating for capacitors
  - as a time from \(-\infty\) to time \(t\) of the current that has gone through the capacitor
- is it possible to express the impulse as a linear combo of the unit steps
- \(\lambda[n] = u[n] - u[n-1]\)
CT Unit STEP
0 all the way up to 0 and then it is 1
\(u(t) = \{0 \text{ at } t < 0 \text{ and } 1 \text{ at } t \geq 0\}\)