Course Learning Objectives and Outcomes

Course Learning Objectives and Outcomes:

This course trains students for an intermediate level of fluency with signals and systems in both continuous time and discrete time, in preparation for more advanced subjects in digital signal processing (including audio, image and video processing), communication theory, and system theory, control, and robotics. 

Upon successful completion, a student should:

• Be able to classify systems based on their properties: in particular, to understand and exploit the implications of linearity, time-invariance, causality, memory, and bounded-input, bounded-out (BIBO) stability.
• Determine Fourier transforms for continuous-time and discrete-time signals (or impulse-response functions), and understand how to interpret and plot Fourier transform magnitude and phase functions.
• Understand the sampling theorem and how it links continuous-time signals to discrete-time signals.  In particular, know how to derive the sampling theorem from first principles—from the basic properties of the Fourier transform; how the spectrum of a sampled signal relates to the spectrum of the original signal; how to use the sampling theorem to understand aliasing phenomena in the real-world, and how to reduce or prevent aliasing; and how to perform discrete-time processing of continuous-time signals, and vice versa, using A-to-D and D-to-A converters.
• Understand the need to define two new transforms—the Laplace and Z transforms—to treat a class of signals broader than what the Fourier transform can handle.
• Understand the combined implications of linearity and time invariance in the Laplace and Z transform domains.  In particular, know how to represent the response of an LTI system to a more general form of complex exponential—e^st in continuous time or zⁿ in discrete-time—and understand that complex exponentials are eigenfunctions of LTI systems; use the Laplace transform to determine the transfer function of a continuous-time LTI system; solve for a response given the input, system description and initial conditions; and answer questions related to BIBO stability, representations of continuous-time signals and systems; use the Z transform to determine the transfer function of a discrete-time LTI system; solve for a response given the input, system description and initial conditions; and answer questions related to BIBO stability, including the central role of the unit circle in the transform-domain representations of discrete-time signals and systems; represent an LTI system by its transfer function; determine the input-output behavior of an LTI system entirely in the transform domain, using relationships between time-domain and the frequency-domain (e.g., convolution in the time domain corresponds to multiplication in the frequency domain); understand the conditions under which the transfer function of a system (or the Laplace or Z transform of a signal) is rational, and know that a continuous-time LTI system with a rational transfer function can be represented by a linear, constant-coefficient differential equation; and a discrete-time LTI system with a rational transfer function can be represented by a linear, constant-coefficient difference equation.
• Understand the relationships among the various representations of LTI systems—linear constant-coefficient difference or differential equation, frequency response, transfer function, and impulse response—and infer one representation from another (e.g., determine the impulse response from the difference equation, etc.).
• Understand the conditions for a time-domain function to have a Fourier transform, and know how to relate the Fourier transform to its Laplace or Z transform.
• Understand the various properties of the four Fourier transforms, the Laplace transform, and the Z transform—including time-shift, modulation (frequency shift), duality, symmetry and anti-symmetry—and exploit them to analyze and design signals and systems.