Signals and Systems  

• Signal representation
  • Complex signal representation
  • Signal space representation: vector space, functional space, orthogonal expansions of signals
  • Phasor
• Use sinusoidal signals to analyze system performance (e.g., frequency response).  
  • Why are sinusoidal signals important?  Because any signal can be decomposed into a mixture of sinusoids as long as the Fourier transform of the signal exists.   
  • Why do we use sinusoidal signals to analyze system performance?  Because 1) sinusoidal input to a linear system generate sinusoidal response of the same frequency; 2) any signal can be decomposed into a mixture of sinusoids; 3) we are mainly interested in linear systems.   Actually, we first use an impulse  (composed of sinusoids of all frequencies) to test the system and then take the Fourier transform to obtain the frequency response.
• Represent sinusoidal signals as complex functions of time, i.e., a complex exponential exp(jwt).   
  • Is a complex signal a physical signal or just a mathematical manipulation? What does exp(jwt) mean?  
  • (Mathematical explanation) Complex exponentials are convenient representations of sinusoidal signals.   They are just a form of representation of signals.   
  • (Intuitive explanation) A physical interpretation of a complex exponential exp(jwt)= cos(wt) + j*sin(wt), is that the complex exponential is a mixture of two orthogonal signals, i.e., the inphase component cos(wt) and the quadrature component sin(wt). Here, the `j' represents the orthogonal relationship between cos(wt) and sin(wt).  In communication systems, we can use coherent demodulation to separate the inphase and quadrature components.
• How do you analyze a signal?  Analyze a signal in terms of spectrum.  
  • What does a negative frequency mean? Comparing exp(-jwt)= cos(wt) - j*sin(wt) with exp(jwt)= cos(wt) + j*sin(wt), we see the complex exponential with a negative frequency characterizes the inphase and quadrature components of the signal at the frequency w but with different phase relationship between inphase and quadrature.   In the case that we have exponentials with symmetric negative and positive frequencies, we obtain real signal, e.g.,  exp(jwt)+exp(-jwt) = 2*cos(wt).
  • So negative frequency only has mathematical meaning but does not have physical interpretation.
• Why no signal can be both time-limited and bandwidth-limited?
  • (Mathematical proof) Use windowing technique.   Any time-limited signal x(t) is the product of itself x(t) and a rectangular window g(t).  The Fourier transform of a rectangular window is a sinc function of frequency, denoted by sinc(w).   So the Fourier transform X(jw) of x(t) is the convolution of X(jw) and sinc(w).   Since sinc(w) has infinite duration in freqency domain, X(jw) convolved with sinc(w) also has infinite horizon in freqency domain.   Therefore, a time-limited signal x(t) has infinite spectrum.   It cannot be bandwidth-limited.   The converse can be proved similarly.
  • (Intuitive explanation) A time-limited signal is aperiodic because its duration is finite.  So its sinusoidal expansion (i.e., Fourier transform), which consists of infinite-duration periodic sinusoids, should have infinite (why not finite?) number of sinusoids to cancel out each other outside the duration of the time-limited signal.   So a time-limited signal has infinite spectrum.   Similarly, a bandwidth-limited signal has infinite duration.
• How do you analyze a system?  
  • A system is characterized by its input-output relationship.  So we have to first apply a signal to the input of the system; then observe the output signal.
  • Transient analysis vs. steady state analysis
  • Time response vs. frequency response.   Time response could be impulse response, step-input response, ramp-input response, parabola-input response, etc.
  • Continuous time system vs. discrete time system.  For continuous time system, use Fourier transform/Laplace transform.  For discrete time system, use z-transform. 
• Deterministic signal vs. Statistical signal
  • Spectrogram vs. Periodogram
  • Power Spectrum, Bispectrum, trispectrum, polyspectrum
• Linear system vs. nonlinear system
  • Linear system:
    • Measure once, predict forever
    • That is, we can measure the output of the linear system, given the input of impulse \delta(t); we obtain the impulse response h(t).   Then, for an arbitrary input x(t), we do not need to run the system and measure the output.  We can predict the output precisely, which is y(t)= x(t) * h(t), where * is convolution.
  • Nonlinear system:
    • Always measure, no prediction
    • That is, for each different input, we need to run the nonlinear system and measure its output; generally, there does not exist a formula for predicting the output, given an input and impulse response or something equivalent.
• Sampling in time domain leads to periodicity in frequency domain, while sampling in frequency domain results in periodicity in time domain.
• Zero order hold (ZOH) and n-th order hold
  • ZOH: first sample the continuous-time signal and then hold the sampled signal for a duration equal to the sampling interval.   It is used to hold the sampled signal in order to process/transmit/reconstruct the discrete-time signal.  The impulse response of ZOH excluding the sampling part is a rectangle.
  • First order hold (FOH): The impulse response of FOH excluding the sampling part is a triangle, i.e., FOH = two cascaded ZOH's.
  • n-th order hold = n cascaded ZOH's.   When n approaches infinity, the impulse response approaches a Gaussian probability density function.
• Inverse relationship between time and frequency (Heisenberg uncertainty principle)
  • The smaller the granularity in time, the larger the granularity in frequency.
  • The larger the granularity in time, the smaller the granularity in frequency.
• ISI is signal overlap in time while aliasing is signal overlap in frequency.
  • Intersymbol interference (ISI) is caused by non-ideality of the channel.  An ideal channel has constant amplitude frequency response and linear phase frequency response.  The phenomenon of ISI is that signals that are originally in different time slot, overlap at the receiver.  
  • Aliasing is caused by under-sampling a band-limited signal below its Nyquist rate (when converting continuous-time signal to discrete-time signal or down-sampling a discrete-time signal).   This leads to the overlap of shifted signal spectrums.   Note that sampling a band-limited signal results in periodically shifted signal spectrums.
• Singularity functions
  • Also called generalized functions, distribution, and ideal function.
  • Dirac delta function,  (continuous time) unit impulse function. 
    • Cross-reference: Kronecker delta function is the discrete version of the delta function.  It is also called (discrete time) unit impulse/sample (function).
• The system transfer function H(s) or H(z) of an LTI system (continuous/discrete time) must be rational, i.e., H(s)=N(s)/D(s), where N(s) and D(s) are polynomials in s.

• Comparisons of various transformations
  • Continuous-Time Fourier Series (CTFS) vs. Continuous-Time Fourier Transform (CTFT)
  • Discrete-Time Fourier Series (DTFS) vs. Discrete-Time Fourier Transform (DTFT) 
  • Discrete Fourier Series (DFS) vs. Discrete Fourier Transform (DFT)
    • DTFS and DFS are the same.
    • The formulae for DFT and DFS are the same, except that the time sequence x(n) in DFS is periodic and x(n) in DFT is aperiodic.
  • Short-Time Fourier Transform (STFT) vs. Wavelet
  • CTFS, DTFS, DFS
  • CTFT, DTFT, DFT
  • FT, Laplace Transform, z-Transform
  • z-Transform vs. DTFT
  • Discrete Cosine Transform (DCT), Discrete Sine Transform (DST), Karhunen Loeve Transform
• When to use a specific transform?
  • Fourier transform vs. Laplace transform
  • Fourier series vs. Fourier transform
• Why is Fourier transform so useful in analysis and design of systems?
  • Because, for any function in L^1[-\infty,+\infty]  (or any finite energy signal), its Fourier transform (in the sense of Lebesgue integral) must exist.
    • Proof:  \int |f(t)*exp(-jwt)| dt <=  \int |f(t)|*|exp(-jwt)| dt <= \int |f(t)| dt < \infty.   Then the Fourier transform \int f(t)*exp(-jwt) dt must exist since \int |f(t)*exp(-jwt)| dt <\infty.
    • Finite-power periodic signals, such as sin(wt), which is not in L^1[-\infty,+\infty], also have Fourier transform representations, in addition to Fourier series representation.   
  • Fourier transform is an operator, which maps a function f in L^1(R^n) to another function \hat{f} in C^0(R^n).
  • Sufficient condition for existence of Fourier transform of a function f:
    • f \in L^1[-\infty,+\infty]
  • Necessary condition for existence of Fourier transform of a function: L^2(R) Hilbert space with inner product defined?
  • A Hilbert space always admits an orthonormal basis, so that every function f in a Hilbert space admits a Fourier series representation. What about Fourier transform? 
• Fourier series: 
  • Every function in a separable Hilbert space has a unique generalized Fourier-series representation w.r.t. the corresponding orthonormal set.
  • Examples of a separable Hilbert space: L^2[a,b] with an inner product defined.
• Parseval's relation: energy/power conservation law
  • For a finite-energy signal, total energy spread over time = total energy spread over frequency
    • So we can obtain total energy of a signal by either integrating/summing up the energy at each time instant or integrating/summing up the energy density at each frequency.
  • average power in time domain = average power spread over frequency
  • sum/integral of energy in time domain = sum/integral of energy in frequency domain
  • average power in time domain = energy/average power in frequency domain
• Partial fraction decomposition/expansion/representation of rational system transfer function H(s)
  • It makes inverse Laplace transform easy.  E.g., given a system characterized by a differential equation, first use Laplace transform to convert the differential equation to an algebraic equation; then solve the equation for H(s); if impulse response (in time domain) is of concern, compute the inverse Laplace transform L^{-1} of H(s).  A simple method of computing L^{-1}(H(s)) is by decomposing H(s) into simple functions, the inverse Laplace transform of which can be easily obtained.
  • It reduces a high-order system into cascade/parallel of first order and/or second-order systems.
  • Partial fractions correspond to first order or second-order systems.

Sensors

Distance

• Linear Potentiometer
• Proximity Detector
• Variable Inductor
• Polaroid Sonar & PASCO Sonar
• PSD
• LVDT

Angular Position

• Optical Encoder
• Conductive Plastic Potentiometer
• Potentiometer
• Optical Interrupter
• Tilt
• Gyro

Force

• Load Cell
• PASCO Force Sensor
• Pressure Sensor
• Strain Gauge

Light

• Photo Resistors
• Solar Cells
• Pressure Sensor

Proximity

• Hall Effect

Speed

• Tachometer
• Accelerometer

Temperature

• Thermometer
• Thermal Couple
• RTD