ECE 301 Signals and Systems

Extremely Short Summary of Each Week’s Materials

In this document we summarize about 30% of the weekly materials. They are not meant to be used as a substitution of the weekly lectures. Instead, it should be used to let you know which part (roughly) of the weekly material is relatively more important the other parts. If you only study the subjects listed here, you won’t be able to get a full picture of this course and it would be reflected in your exam scores. However, if you are really short of time, the list here provides you a good starting point how to review/study the weekly materials.

=== Start of the summary ===

Week1 Key Information

· Linearity

o Know how to use the two-configuration method to justify the linearity of a given input-output relation.

· Definition of continuous signal and discrete signal

· Definition of energy and power

o Know how to compute energy and instantaneous/average power for a given signal.

Week2 Key Information

· Algebra of signals

o Specifically, time shifting, time reversal and time scaling. They are basic mathematical techniques to generate a new signal by manipulating a given signal. Practice the composite transformation example given in the lecture notes.

· Periodic signals, even & odd signals.

o Understand the definitions and how to find the period of a signal.

· CT complex exponential signals with detailed illustration

o Know how to find the magnitude, phase and the fundamental frequency/period of a CT complex exponential.

Week 3 Key Information

· DT Complex exponential signals

o Practice finding the fundamental frequency/period. Examples in P036-037.

· The concept of harmonically related complex exponentials.

o Understand it. It is the important concept for building up Fourier transforms.

Week 4 Key Information

· The unit step signal and the unit impulse signal, impulse response and step response

o Know the definitions and the sampling property of the unit impulse signal.

· System properties

o Distinguish between memoryless and causal. A memoryless system must be causal, but the reversed is not true. Distinguish between time-invariance and linearity, they are two very different concepts.

Week 5 Key Information

· Convolution

o Understand how impulse response is developed by using the LTI property and sampling property of the unit impulse signal.

o Practice computing convolution.

o Practice using the impulse response to verify stability, memoryless, causality, and invertibility for a LTI system.

Week 6 Key Information

· Inputting complex exponentials to LTI systems

o It is an alternative way for finding output when inputting a complex exponential/sinusoidal signal into a LTI system. Practice it.

· CTFS: the analysis equation and synthesis equation

o Understand the meaning for these two equations: The synthesis equation represents an arbitrary periodic CT signal into a weighted sum of HRCEs. The weights are the Fourier series coefficients. They are found by the analysis equation.

· Two methods to compute Fourier series for CT periodic signal

o Method 1: By inspection for signals which can be easily represented using complex exponentials;

o Method 2: Directly by analysis equation. Practice on some basic signals, such as impulse train and rectangle wave signal.

Week 7 Key Information

· The properties of CTFS

o They are summarized in Table3.1. The ones frequently used are time-shifting property, differentiation property, multiplication property and Parseval’s relationship.

o Get familiar with these properties and be able to recognize them in a question.

Key Strategy for MT3 and Final Exam: The property tables (Tables 3.1, 3.2, 4.1, 4.2, 5.1, 5.2) are very helpful! These tables summarize a very good amount of the CTFS, DTFS, CTFT, and DTFT properties covered in this course. It will serve you well (especially on MT3) to study these tables and recognize when the properties given can help you solve a problem.

Week 8 Key Information

· All properties discussed in Week 8 are summarized and made available in Tables 3.1 and 3.2.

o Recognizing and applying the properties from these tables is perhaps more important than the derivations of the properties themselves.

· All properties are relevant, but very important properties include: Linearity, Time-shift, Multiplication, Parseval’s, Convolution

Week 9 Key Information

· Definition of the CT Fourier Transform and Inverse CTFT (9.1)

· Understand the “sinc” function/rectangular function FT pair (introduced in 9.2, listed in Tables 4.2 and 5.2)

· Recognize where CTFT properties appear in Table 4.1 and when to apply them

Week 10 Key Information

· Understand that convolution in the time domain implied multiplication in the frequency domain, and vice versa; these properties are listed in Table 4.1 (Topics 10.1-10.2)

· Be able to recognize and apply FT pairs from Table 4.2

Week 11 Key Information

· Understand the process of AM Frequency Division Multiplexing (covered in 11.3, but building upon 11.2 and 11.1)

o Understand both the “equations” side of the process and the graphical representation of the process (primarily the graphical representations in the Fourier/frequency domain)

· Understand the process of SSB modulation/demodulation (very similar to FDM, with a new principle)

o Again, understand bot the “equations” side and the graphical side

o Understand why we might want to use SSB modulation

Week 12 Key Information

· Definition of the DT Fourier Transform and Inverse DTFT (12.1)

o The DTFT must be periodic with period 2*pi

· Recognize that most properties of CTFT have parallel similar properties for DTFT (12.2)

o Understand convolution and multiplication properties (12.2 & 12.3)

o Know what properties appear on Table 5.1 and how to apply them

o Know what DTFT pairs are listed on Table 5.2 and how to apply them

Week 13 Key Information

· Understand the ideas of why we would do sampling (13.1)

· Understand what is impulse train sampling and what effect it has on both time and frequency domains (13.2)

· Understand Nyquist sampling theorem (13.2) – At what frequency must we sample a signal in order to effectively capture all information in the signal?

· Band-limited interpolation (13.3)

Week 14 Key Information

· Discrete-time signal processing (14.1)

· Definitions of Z-transform (14.3) and region of convergence (14.4)

· Table 10.1 and Table 10.2