# ECE265: MIDTERM EXAM 1 STUDY GUIDE

Some of the more important things that you should know for the exam include
those items listed below.  These items have been listed in the approximate
order in which they were covered in the course, grouped approximately by
chapter/appendix of the textbook.

## Preamble

### Remarks on Terminology/Notation

To avoid having to write "function/sequence" in many places, the
more generic term "signal" is often used (to mean either a function
or sequence).

### Formula Sheet

For some exams, the exam paper includes a formula sheet with select
formulas, tables, and so on.  If an exam includes a formula sheet,
a draft version of the formula sheet will always be made available in
advance of the exam so that students will know exactly what information
is provided on this formula sheet.  A formula sheet is intended to serve
two primary purposes:

 1. to reduce some of the memorization work required for exams; and
 2. to limit the damage if a student has a sudden brain freeze and
    forgets a very basic formula that they really should have
    memorized (such as the Euler formulas).

Sometimes conditions/constraints on formulas may be omitted from the
formula sheet.  This may be done either to save space or because the
conditions/constraints are expected to be known by the student.  So,
it is important to be aware of any such conditions/constraints for
items on the formula sheet.

## Unit 1: Signals and Systems Basics

### Chapter 1 (Introduction)

- know basic signals/systems terminology and notation (e.g., definitions of
  signal, system, continuous-time, discrete-time, analog, digital, 1D,
  multidimensional, multi-input, single-input, etc.)

### Appendix A (Complex Analysis)

- know Euler's formula
  (be able to express sin and cos in terms of exponentials and vice versa)
- know basics of complex numbers (e.g., Cartesian/polar form, real/imaginary
  part, magnitude, argument, principal argument, addition, multiplication,
  division, conjugation)
- know definition of polynomial and rational function
- know definition of poles/zeros, order of pole/zero, how to find
  and plot poles/zeros of a rational/polynomial function
- know basic properties of polynomial and rational functions
  (e.g., where they are continuous, differentiable, and analytic)

### Chapter 2 (Preliminaries)

- know what rational number is
- know notation used for sets of integers, real numbers, complex
  numbers, rational numbers
- know abbreviated notation for ranges of integers and ranges of real numbers
- know basic terminology such as mapping, domain, codomain,
  function, sequence, system operator
- understand the difference between function/sequence and transform
- know operator grouping (i.e., right to left)
- know definition of even and odd signals (e.g., x(-t) = x(t), etc.)
- know definition of conjugate symmetric signals
- know definition of periodic signal
- know definition of period, frequency, fundamental period, fundamental
  frequency
- understand meaning of various notation that relates to
  function, sequences, and system operators
- understand decibel (dB) measure

### Chapter 3 (CT Signals and Systems) and Chapter 8 (DT Signals and Systems)

- understand transformations of independent/dependent variable
- know definitions of these independent-variable transformations:
    - time shifting
    - time reversal
    - time scaling
    - downsampling
- understand what each independent-variable transformation does to a signal
- be able to plot (independent-variable) transformed versions of signals,
  where transformation may be composition of multiple (independent-variable)
  transformations
- understand basic commutative properties of these independent-variable
  transformations (i.e., most do not commute so order matters)
- be able to relate expressions like x(at-b) or x(an-b) to
  independent-variable transformations (where t and n correspond to
  independent variable)
- know the meaning of amplitude scaling, amplitude shifting

- know definition of least common multiple (LCM) and how to compute it
- know the properties of even/odd signals for addition/multiplication
- know formulas for determining even and odd part of signal
- know the properties of periodic signals for addition
  (i.e., know how to determine if the sum of two or more periodic
  signals is periodic)

- know definitions of left-sided, right-sided, two-sided, finite-duration,
  time-limited, causal, anticausal signals
- know definition of bounded signal
- be familiar with 1-norm and (square of) 2-norm of signals
- know definition of signal energy (i.e., square of 2-norm)

- know definitions of floor and ceiling functions and notation used for them
- know definitions of elementary signals, such as:
    - real sinusoidal functions and sequences
    - complex sinusoidal functions and sequences
    - real exponential functions and sequences
    - complex exponential functions and sequences
    - unit-step function and sequence
    - signum function
    - rectangular function
    - CT and DT unit rectangular pulses
    - (Dirac) delta function and (Kronecker) delta sequence
    - sinc function
- understand quirky behavior in oscillation rate for real/complex sinusoidal
  sequences (e.g., oscillation rate has upper limit)
- be able to determine fundamental period (and frequency) of
  CT/DT real/complex sinusoidal signal
- know definition of (Dirac) delta function and its properties (e.g.,
  equivalence, sifting, scaling, and even)
- know definition of (Kronecker) delta sequence and its properties (e.g.,
  equivalence, sifting, even)
- understand how to collapse multi-case formula for signal into single
  case using unit-step signals

- understand basic idea of block diagram representation of system
- know definitions of series/cascade and parallel interconnection
- know definitions of (CT/DT) system properties:
    - memoryless and memory
    - causal
    - invertible
    - BIBO stable
    - time invariant (TI)
    - linear
- be able to precisely state the conditions that must be satisfied
  for each property
- be able to determine if a system has each one of these properties
- be able to use linearity and time invariance to find response of LTI system
  to linear combination of time-shifted inputs
- know definition of eigensignal (i.e., eigenfunction/eigensequence) of
  CT/DT system (and corresponding eigenvalue) and understand why
  eigensignals are important

- **Omitted content.**
  Some content in the textbook was not covered in the lectures and therefore
  students are not responsible for this content.  This content is identified
  below, classified by textbook section:
    - in Section 3.4.5 "Signal Energy and Power":
        - the definition of signal power was not covered
        - the definition of energy was covered, however
    - Section 3.5.7 "Indicator Function":
        - the indicator function notation was not formally covered
          although it does appear on some lecture slides that were covered
    - Section 3.5.8 "Triangular Function":
        - the definition of the triangular function was not covered
          (if its definition is ever needed, it will be provided)
    - Section 3.5.10 "Aliased Sinc Function":
        - the definition of the aliased sinc function was not covered
          (if its definition is ever needed, it will be provided)
    - in Section 3.5.11 "Rounding-Related Functions":
        - the definitions of the floor and ceiling functions were covered
        - the numerous identities listed involving the floor and/or ceiling
          functions were not covered
        - the definition of the remainder function (mod) was not formally
          introduced (if the mod function is ever needed, its formula will
          be provided)
    - Section 8.2.4 "Upsampling":
        - upsampling was not covered
    - in Section 3.8.6 "Linearity" and
      in Section 8.7.6 "Linearity":
        - additivity and homogeneity were not formally defined; instead
          only linearity was defined (which means additive and homogeneous)

## Unit 2, Part 1: LTI Systems

### Part 1 of: Chapter 4 (CT LTI Systems) and Chapter 9 (DT LTI Systems)

- definition of CT and DT convolution
- properties of CT and DT convolution (e.g., distributive, associative,
  commutative, convolutional identity)
- how to compute CT and DT convolution using graphical method
- how to compute DT convolution using tabular method
- be able to manipulate equations/formulas involving CT/DT convolution
  (e.g., using change of variable to express given integral/sum in terms
  of convolution)

- **Omitted content.**
  Some content in the textbook was not covered in the lectures and therefore
  students are not responsible for this content.  This content is identified
  below, classified by textbook section:
    - Section 4.4 "Periodic Convolution":
        - periodic convolution was not covered
    - Section 9.4 "Periodic Convolution":
        - periodic convolution was not covered

## MATLAB 

### Appendix D (MATLAB)

- do not need to know everything in Appendix D
- only more basic coverage of MATLAB

- basic language syntax
- arrays (i.e., vectors/matrices), array subscripting
- start:step:end syntax
- arithmetic operators (e.g., +, -, *, /, ^, .*, ./, .^)
- relational operators (e.g., ==, ~=, <, <=, >, >=)
- logical operators (e.g., &, |, ~, &&, ||)
- element-wise operations
- basic looping constructs (e.g., for, while)
- basic conditional constructs (e.g., if-then-else)
- user-defined functions
- know some very basic non-math functions (e.g., size, length, plot) and
  basic math functions (e.g., real, imag, abs, angle, sin, cos, exp)
- how to avoid unnecessary looping and conditional-code execution constructs