Signals, Systems, and Controls
* The unit impulse function δ(t) has area equal to _________________ in an infinitesimally short interval.
1
What is a harmonic series?
A function for which the frequency of the higher-frequency component [nω] is an integer multiple of the lower one; the ratio of the two frequencies/periods is equal to the integer n or 1/n
* A signal consisting of a sum of two or more periodic functions will always be periodic. T/F
False
* If a signal x(t) is Laplace transformable, then it must also be Fourier transformable and thus its Fourier transform also exists. T/F
False
* The low frequency limit on the horizontal axis of a Bode plot is always ω = 0 rad/s.
False
What is important to remember when finding phase angles, amigos?
Four quadrant arctangent. Do not forget. Draw the real/imaginary plot pls
What is a transfer function?
H(s) developed using Laplace transform properties
Generalized Fourier transform
Let x(t) be a function defined on −∞ < t < ∞. If there exists some X (ω), not necessarily a continuous function but for which the inverse transform exists, i.e., F^(−1) {X (ω )} = x(t) , then X (ω) is the generalized Fourier transform of x(t)
* Are initial conditions used with FRFs or Fourier transforms?
No, but they are used for Laplace transforms (that result in transfer functions)
What is the definition of a Laplace transform?
X (s) =∫x(t) e^(−st) dt evaluated from 0 to inf
What is the general form of the Fourier transform?
X(ω) = ∫x(t) e^(−jωt) dt evaluated from -inf to inf = F {x(t)}
What is causality?
a property which means that a system's response y(t) at any time t can't be a function of any future inputs; it can only depend on inputs occurring in the past, up to the present time
The constitutive law governing a mass (f = mx(doubledot)) requires a(n) ________________ displacement, while the laws governing a spring or damper are based upon ________________ displacements.
absolute/inertial; relative
What are signals?
all time-dependent functions, both inputs and outputs
For the Fourier series, if all of the sine terms are 0, what type of function do we have?
an even function where the series is a cosine series
For the Fourier series, if all of the cosine terms are 0, what type of function do we have?
an odd function where the series is a sine series
What is a periodic function?
any function which satisfies: x(t + T ) = x(t) , −∞ < t < ∞ for some value of T (the period)
For a Fourier series, the value of Ck at a particular value of the index k is known as the ________________ of the signal at that particular frequency
component
For a system with impulse response h(t) ≠ 1 and input u(t), what is the general integral equation which gives the system output y(t)?
convolution integral (zero initial condition response): y(t )=∫u(τ)h(t−τ)dτ evaluated from 0 to t
Convolution property of Fourier transform
convolution of two functions in the time domain is equivalent to multiplication of the transforms of the two functions in the frequency domain
What is the difference between design and control of a system?
design: the system input is viewed as fixed and the physical system is modified to produce acceptable output control: the physical system is viewed as fixed and the input is modified to produce acceptable output
What is a homogeneous differential equation?
differential equation with no input [u(t) = 0]
* Gibbs phenomena occur in the Fourier series representation of a signal when the original signal contains ______________________________.
discontinuities
What does it mean to model a system?
find the differential equation(s) which govern the response
Other names for the particular solution of an ODE?
forced solution or steady-state solution
What is another name for a system input?
forcing function
Paley-Wiener criterion???
from Chap. 6: Appendix C, a necessary and sufficient condition for ∣H(ω)∣ to be the magnitude frequency response of a causal system impulse response h(t) is ∫∣ln ∣H(ω)∣∣/(1 +ω^2) dω < ∞ evaluated from -inf to inf
The fundamental frequency (ω0) of two functions (with ω1 and ω2) added together will be equal to the ________________________________ of the two frequencies ω1 and ω2.
greatest common divisor
* The guitar string and the piano wire, vibrating at the same fundamental frequency, sound different from one another due to _______________________
harmonic content
* A system's frequency response function (FRF) is the Fourier transform of its ______________________.
impulse response
* The (generalized) Fourier transform of the signal x(t) in the previous question (sum of two sinusoids at frequencies ω1 and ω2) will contain what type of function(s)?
impulse δ(t)
Coupled variables are _____________ but are related through _________________.
independent; a set of differential equations
The Fourier series is an _____________ _____________ series.
infinite, harmonic
All dynamic systems _____________ the input in some form.
integrate
* If an aperiodic (no finite period) signal x(t) consists of a sum of two sinusoids at frequencies ω1 and ω2, then the ratio ω2/ω1 is ______________.
irrational
The fundamental period (T0) of two functions (with T1 and T2) added together will be equal to the ________________________________ of the two periods T1 and T2.
least common multiple
Any sum of a countable number of sinusoids on −∞ < t < ∞ , whether the sum is periodic or not, will have a ___________________
line spectrum
What are inhomogeneous or forced differential equations?
linear ODEs with inputs
How can you identify dynamic (or inertia) coupling?
mass matrix has non-zero off-diagonal elements
Sinusoids at different frequencies, when added together, can create a wide range of different waveforms that ________________________ periodic.
may or may not be
An output must normally be a physical variable which is...
measurable
MDOF problems are described by _______________, ________________ differential equations
multiple, coupled
Some obscure sentences in the notes that were italicized/underlined: The effect of an impulse input at t = 0 is equivalent to a _____________________________. As a result, a system's impulse response is qualitatively the same as the _______________________________.
non-zero initial condition; homogeneous solution
What is Gibbs phenomena?
occurs where there is discontinuity of a signal, the oscillations seen in a truncated Fourier Series; with more terms in the summation, the plot will converge to points of discontinuity
Ordinary frequency vs. angular/radian frequency?
ordinary --> f; angular/radian --> ω
* For a signal x(t), a plot of x(t) vs. x(t) (i.e., a phase plot or Lissajous figure) is used to check graphically for ___________________________.
periodicity
For a linear ODE, the characteristic equation will always be a...
polynomial
If a sum of sinusoids is not periodic, it lies in a special subset of aperiodic functions known as ______________________
quasi-periodic
What is the state variable (or state space) form?
representing a n-th order system's differential equation(s) as a set of n coupled first-order differential equations; variables used to define the model (x1, x2, ..., xn) are called states
What indicates that a system is underdamped when considering a solution to a differential equation?
roots are complex
What is the equation for undamped natural frequency of a mechanical system (because I have forgotten this twice already this semester)?
sqrt(k/m)
Dependent variables exist if there is a _____________ relationship between them.
static/algebraic/kinematic
How can you identify static coupling?
stiffness matrix has non-zero off-diagonal elements
What are linear ODEs governed by?
superposition principle
What is another name for a system output?
system response (or time response)
* What is the necessary condition for the Fourier Transform to equal the Laplace Transform?
the Fourier transform is equal to the Laplace transform for s = jω if the Fourier transform exists in the regular (i.e., continuous) sense
* For a periodic signal containing discontinuities, the Fourier series representation of the signal converges to ______________________ at the discontinuities.
the average of the left and right hand limits
What does it mean to normalize a differential equation?
the coefficient of the highest order term is 1
What is the fundamental frequency of a harmonic series?
the harmonic series x(t) will be a periodic function and its frequency is equal to the frequency of the fundamental (lowest frequency) component
What is rolloff?
the high-frequency slope of the Bode magnitude plot (aka that thing we did in lab 2 where we calculated the slope to be -20 dB/decade for 1st order and -40 dB/decade for 2nd order)
If the FRF H(ω) is dimensionless, then for a sinusoidal input having ∣H(ω0)∣ > 1 means...
the input signal is amplified at that frequency
If the FRF H(ω) is dimensionless, then for a sinusoidal input having ∣H(ω0)∣ < 1 means...
the input signal is attenuated at that frequency
What is the sifting property of the unit impulse function?
the integral ∫ x(t)δ (t−t0) dt = x(t0) from -inf to inf also works over any subinterval from a to b
What is an impulse response?
the output of a system to which a unit impulse is applied at the input (with zero initial conditions)
What does it mean for two frequencies ω1 and ω2 to be incommensurate?
the ratio ω2/ω1 is an irrational number and any sum containing two or more such frequencies is aperiodic
What does it mean for something to be a dynamic system?
the response is time-dependent; the input and output are related through one (or more) differential equations
What are harmonics?
the sinusoidal components at frequencies above the fundamental frequency
What is the fundamental period?
the smallest positive value of T for which a function is periodic
The only practical way to find impulse response is to use Laplace transforms. Why?
the unit impulse unit is not differentiable
A time function x(t) and its Fourier transform X(ω) are together referred to as a _______________________
transform pair
What is a decade?
two frequencies are said to be an decade apart on the frequency scale if their ratio ω2 /ω1 is equal to 10
What is an octave?
two frequencies are said to be an octave apart on the frequency scale if their ratio ω2 /ω1 is equal to 2
sinusoid
u(t ) = Acos(ωt) + Bsin(ω)t , −∞ < t < ∞
exponential function
u(t ) = e^(−at), t ≥ 0
unit ramp function
u(t ) = t , t ≥ 0
unit step function
u(t) = 1, t ≥ 0
The Fourier transform H(ω) is undefined for a(n) __________________ system at the resonant frequency
undamped
What is the general form of the inverse Fourier transform?
x(ω) = 1/(2pi) ∫X(ω) e^(jωt) dω evaluated from -inf to inf = F^(-1) {x(t)}
* A magnitude ratio of 0.01 is equal to -40 dB. T/F
True
* A periodic function with period T0 is also periodic with period nT0, where n is any integer, n > 1. T/F
True
* The impulse response of an ideal integrator system is the unit step function. T/F
True
* The magnitude plot of a Fourier transform must be an even function. T/F
True
* The separation between two frequencies at 100 rad/s and 200 rad/s is one octave. T/F
True
* The separation between two frequencies at 20 rad/s and 200 rad/s is one decade. T/F
True
* When using a system's FRF H(ω) to analyze response to sinusoidal inputs, only the particular solution is used; the homogeneous solution is neglected. T/F
True
* sin(ω0 t) and cos(ω0 t) are not Fourier transformable in the regular, continuous sense, but each has a generalized Fourier transform. T/F
True
* The _______________________ conditions give sufficient conditions for existence of the Fourier transform.
Dirichlet
What are the conditions generally used to claim that a Fourier Series exists for a function (exists in the "regular" or "continuous" sense)?
Dirichlet conditions: on any finite interval x(t) is bounded, has a finite number of maxima and minima, and has a finite number of discontinuities; x(t) is absolutely integrable (∫ ∣x(t )∣dt <∞ evaluated from -inf to inf
A sum of two periodic functions with periods T1 and T2 will be periodic only if...
T1/T2 is a rational number (ratio of integers)
What is the magnification factor?
The magnitude function ∣H(ω)∣; the factor by which the amplitude of a sinusoidal input, at a specific frequency, is multiplied at the output
