3.1  To Learn in This Chapter

The skills we aim to develop in this chapter are:

• Learn how system models $H$ are used to characterize the output signal $y$ corresponding to a given input signal $x$ in both continuous-time and discrete-time
• Understand what a linear and time-invariant (LTI) system is and the role of the convolution operation to determine the output signal
• Learn convolution properties and how to calculate the discrete-time convolution via correlation or matrix multiplication
• Calculate the circular (or periodic) convolution between two signals or two spectra
• Distinguish signals and systems, in spite of the signal called impulse response $h(t)$ being used to represent a LTI system
• Characterize a continuous-time LTI system based on its impulse response $h(t)$ or linear constant-coefficient differential equation relating the input $x(t)$ to the output $y(t)$
• Characterize a discrete-time LTI system based on its impulse response $h[n]$ or linear constant-coefficient difference equation (LCCDE) relating the input $x[n]$ to the output $y[n]$
• Learn the concept of system function and frequency response
• For LTI systems, use Laplace and Z transforms to calculate the system function $H(s)=L\{h(t)\}$ and $H(z)=Z\{h[n]\}$ from impulse responses $h(t)$ and $h[n]$ for continuous and discrete-time, respectively. Similarly use these transforms to obtain $H(s)$ and $H(z)$ from the respective linear constant-coefficient differential or difference equations
• For LTI systems, use continuous-time Fourier transform and DTFT to calculate the frequency response $H(\omega )=F\{h(t)\}$ and $H(e^{j\mathrm{\Omega }})=F\{h[n]\}$ from impulse responses $h(t)$ and $h[n]$ for continuous and discrete-time, respectively. Similarly use these transforms to obtain $H(\omega )$ and $H(e^{j\mathrm{\Omega }})$ from the respective linear constant-coefficient differential or difference equations
• Understand that complex exponentials are eigenfunctions of LTI systems
• Design and use analog and digital frequency-selective filters both via software and analytically
• Implement in software a digital filter using its LCDDE equation
• Get familiar with the definitions of key frequencies for dealing with frequency-selective filters: cuttof, natural, etc.
• Interpret the group delay and use it to evaluate the delay imposed by a system
• Learn to predict how a system modifies an input signal based on system properties
• Reinterpret sampling and signal reconstruction, now being able to use the convolution operation in the mathematical model
• Learn how to perfectly reconstruct a band-limited signal when the sampling theorem is obeyed
• Know details about first and second-order systems
• Learn the definitions of bandwidth and quality factor for filters or individual poles
• Know the group delay of a system is the derivative of its frequency response phase, and how useful is to have a system with linear phase (or equivalently, a constant group delay)
• Learn about commercial filters based on technologies such as SAW and ceramics
• Know the concepts of FIR, IIR, AR, MA and ARMA discrete-time systems
• Design analog filters using filter frequency scaling and bandform transformation
• Design IIR filters using matched Z-transform, impulse invariance, backward difference, forward difference and bilinear transformation (also called Tustin’s method)
• Learn how to use prewarping when designing IIR filters with the bilinear transformation
• Design FIR filters using least-squares or windowing, specially focusing on filters with linear phase
• Learn the most import structures to implement FIR and IIR filters (transposed direct II structure, etc)
• Understand the effects of finite precision when the filter coefficients are quantized and roundoff errors occur during the filtering process
• Learn basic concepts of multirate processing, such as up and downsampling