Fourier and Laplace 321
4.2 Objectives
After completing this chapter, the reader should be able toMathematically state the trigonometric and exponential FS
State that sine and cosine are the basic periodic (orthogonal) functions used to
approximate an arbitrary function
State that any arbitrary periodic function can be resolved into harmonic components
Evaluate the coeffi cients of the FS expansion of any arbitrary signal
Compute the FS coeffi cients of a square and sawtooth wave
Improve the partial sums of the FS by the use of a window function
State the FT and the inverse FT (IFT) equations
Understand the importance of the Fourier analysis
Determine the linear and power spectral distribution of arbitrary signals
Demonstrate that any arbitrary time function can be represented in the frequency
domain by two plots referred as the amplitude and phase spectrum plots
Understand the physical implications of a periodic and nonperiodic signal in the
frequency domain
Evaluate the FT of periodic signals
State the concept and conditions for amplitude and delay distortion
Evaluate the percentage of harmonic and total distortion
Defi ne the concept and purpose of an equalizer circuit (to correct distortion)
Use Parseval’s theorem to calculate the power for periodic and nonperiodic time
signals
Understand the concept of energy and power signals
Revisit the concepts of root mean square (RMS), power, and energy in the fre-
quency domain
State the integral defi nition of the direct and inverse LT (ILT)
Use the LT pairs in the analysis of electrical circuits, systems, and signals
Find the LT of simple functions by using tables, as well as the symbolic MATLAB
toolbox
Understand the purpose and power of the LT in the analysis of electrical circuits
Express electrical circuit elements using the s-domain model (Laplace), where ini-
tial stored energy conditions are incorporated in the model (as sources)
Understand the procedure by which integral differential and DEs can be solved by
using the LT
State and be able to apply the initial and fi nal value theorems in the solution of
electrical circuit problems
Identify the effects of waveform symmetry in the computational steps involved in
spectra analysis
State the basic relationships (referred to as properties) between the time- and
frequency domain