Signals and Systems (Adams)

A free book. This one was advised too.

48 flashcards · Mathematics · by lakesare

Name an example of: one-dimensional signal multi-dimensional signal
one-dimensional signal - human voice (fluctuations of air pressure as a function of time) A signal that's a function of one variable.multi-dimensional signal - image (light intensity as a function of horizontal and vertical position) A signal that's a function of multiple vairables.
What's the difference between: discrete-time and continuous-time signal VS digital and analog signal?
discrete-time and continuous-time signal - is about the domain of the signal. digital and analog signal - is about the domain AND the codomain.
What's meant by discrete-time VS continuous-time signal?
Discrete-time signal - a function of discrete variables E.g., a digital image is a discrete-time signal, even though the independent variables (horizontal and vertical position) do not actually correspond to time.Continuous-time signal - a function of continuous variables E.g., a music recording is a continuous-time signal (because it depends on time).
What is the difference between these two expressions?
(f + g)(t) - denotes a number obtained by: compute a function h by adding the functions f and g evaluate h(t) The addition operation is being applied to two functions,f(t) + g(t) - denotes a number obtained by: evaluate f(t), evaluate g(t) add these numbers The addition operation is being applied to two numbers. Although the meanings of these two expressions are subtly different, they are always equal.
Name some function operations that are defined pointwise? Since the addition of functions is defined pointwise (we add two functions by adding their values at correponding pairs of points), this equality always holds.
However, not all operations on functions are defined in a pointwise manner! Some operations fundamentally require that their operands are functions, e.g. the convolution operator.
What do people mean when they write "the function f(t − τ)" in signal theory?
The function g defined by the following rule: g(t) = f(t − τ). f(t − τ) could have 4 meanings: About avoiding ambiguous notation:
There are 4 types of mapping: functions sequences system operators, and transforms.
function - domain is continuous (aka continuous-time signals) sequence - domain is discrete (aka discrete-time signals) Discrete-time signals are referred to as sequence.The n-th element of a sequence x is denoted x(n) or xn.system operators - mapping of functions to functions Mapping used to represent a system, transforms - mapping of functions to functions
What's meant by a system in signal theory? E.g., Linear Time-Invariant (LTI) system.
A mapping from a function to function. E.g., amplification of sound. A signal is a function.
[System sytnax] What does Hx(t) mean? Where x is a function, and t is a constant.
Hx(t) = (Hx)(t) = (H(x))(t)
What are even VS odd functions symmetric about? A function is even when: x(t) = x(-t) for all t. A function is odd when: x(t) = -x(-t) for all t.
even functions are symmetric with respect to the y-axis. Geometrically, an even function or sequence is symmetric with respect to the vertical axis.odd functions are symmetirc with respect to the origin. Geometrically, an odd function or sequence is symmetric with respect to the origin.
When is function x(t) said to be periodic?
If x(t) = x(t + T), then x(t) is periodic with period T Or simply T-periodic.In other words, x(t) repeats in value every T units across the horizontal axis.
What's the frequency and angular frequency of T-periodic function?
A T-periodic function is said to have the frequency - 1/T angular frequency - 2pi/T
Why do we need to define the fundamental period?
Because if a function is periodic with period T, it's also periodic with period kT (for every strictly positive integer k), and we're interested in the smallest value T. E.g., x(t) is periodic with periods T, 2T, 3T, etc.The fundamental period, however, is uniquely determined, and is equal to T.Morever, the frequency corresponding to the fundamental period is called the fundamental frequency (1/T).
How can this system be described as a formula?
x - input functiony - output function
Express the series interconnection in a formula. Series interconnection of the systems H1 and H2
Express the parallel interconnection in a formula. Parallel interconnection of the systems H1 and H2
[System properties] When is the system considered to have memory?
A system has memory when its definition uses not only x(t), but also, e.g., x(t + 1).
[System properties] When is the system considered to be causal?
A system is causal when its definition uses x(t), x(t - 1), etc., but not x(t + 1).
[System properties] When is the system considered to be time invariant?
A system is time invariant if a time shift in the input function x results in an identical time shift in the output function y H(x)(t - t0) = H((t) => x(t - t0))(t) Systems that are equivalent if H is time invariant (i.e. H comutes with St0):a) A system that first time shifts by t0 and then applies H (y = HSt0x)b) A system that first applies H and then time shifts by t0 (y = St0Hx)
[System properties] What's the name of the NOT time-invariant (TI) system? Also known as shift invariant (SI)
Time varying. Also known as shift varying.
[System properties] When is the system considered to be linear?
A system if linear if it satisfies the superposition property: Systems that are equivalent if H is linear:a) y = H(a1x1 + a2x2)b) y = a1H(x1) + a2H(x2)
[System properties] What 2 properties do LTI systems have?
Linear Time invariant
[Convolution] What does the convolution operator ∗: accept, and return? In the context of LTI systems, a mathematical operation known as convolution turns out to be particularly important.
The convolution operator: accepts - 2 functions, and returns - function.
[Convolution] Where could the brackets be in the x∗h(t) expression.
(x∗h)(t)
[Sum of periodic functions] If x1 is periodic, and x2 is periodic - in what case is y = x1 + x2 periodic?
y = x1 + x2 is periodic iff T1/T2 is a rational number.
[Sum of periodic functions] How to find the period T of y = x1 + x2 (provided it's periodic).
T = lcm(T1, T2) y = x1 + x2 is periodic iff T1/T2 is a rational number. The lcm(a, b) is usually defined between 2 integers, but it can also be defined between 2 real numbers.
[Sum of periodic functions] How would you find the period T of y = x1 + x2?
Find T1, the period of x1 Find T2, the period of x2 Check whether T1/T2 is rational (then y is periodic) Find T = lcm(T1, T2)
[Decomposition of function into even and odd parts] There's only one way to decompose (any) function x into the sum: x = even function + odd function. Which is it?
xe is called the even part of x (denoted Even{x})xo is called the odd part of x (denoted Odd{x}) A proof that this decomposition is unique
[Signal transformations] What's time shifting?
y = (t) ⇒ x(t − b) a) xb) The result of applying a time-shifting transformation to x with a shift of 1
[Signal transformations] What's time compression/expansion/scaling? Time compression, time expansion, time scaling
y = (t) ⇒ x(at)
[Signal transformations] What's time reversal?
y = (t) ⇒ x(−t) a) xb) the result of applying a time-reversal transformation to x
[Signal transformations] What's amplitude shifting?
y = (t) ⇒ x(t) + b
[Signal transformations] What's amplitude scaling?
y = (t) ⇒ ax(t)
What's the support of a function?
Interval where f(x) ≠ 0. Subset of the domain containing the elements which are not mapped to zero.
When is the function left-sided?
When it's f(x) = 0 starting from some t0. Left-sided
When is a function called finite-duration function? Also know as time-limited function.
When it's both left-sided and right-sided. By analogy, a diagonal matrix is a matrix that's both upper-triangular and lower-triangular. A finite-duration function.
Bounded VS time-limited function?
Bounded - limited vertically Time-limited - limited horizontally Also known as finite-duration function.
What's energy signal?
A function with finite energy. As a matter of terminology, a function x with finite energy is said to be an energy signal.
[Elementary functions] Unit-step function VS signum function?
Unit-step function (aka Heaviside function)Signum function (aka sign function) - a mathematical function that extracts the sign of a real number.
[Elementary functions] How does the rectangular function look?
[Extracting part of a fuction with a rectangular pulse] How would you extract 1 period from x using the rectangular function v?
We wanat to multiply x by a function that is 1 over [-T/2; T/2] and 0 everywhere else, v: v * x.
[Elementary functions] What does the indicator function χ[−1/2, 1/2) represent?
The rectangular function.
[Elementary functions] What does the indicator function χ[0, ∞) represent?
The unit-step function. Also known as Heaviside function.
[Elementary functions] What does the sinc(x) function represent?
The cardinal sine function.
[Elementary functions] What's the formula for the normalized sinc(t) function? Unnormalized sinc(t)
[Elementary functions] Name 1 example of a generalized function. In mathematics, generalized functions are objects extending the notion of functions.Generalized functions are especially useful in making discontinuous functions more like smooth functions.
Unit-impulse function (Dirac delta function, delta function) δ δ is defined by 2 properties:From these properies, we can see that δ is defined everywhere apart from t=0, where δ is undefined. Technically, δ is not a function in the ordinary sense.
[Elementary functions] How would you express the rectangular function rect(t) using the unit-step function u(t)?
[Convolution What would the result of computing x∗h *look like* as a formula?
A formula that only depends on t. (No integral, no tau!)