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Write a function file that takes a domain vector t, a scalar amplitude A, and a scalar period T as inputs, and outputs a vector 5 that evaluates the ideal square wave given in Eq.1. Your function file header should match the following function S = SquareWave(t.A.T)

Write a function file that takes a domain vector t, a scalar amplitude A, and a scalar period T as inputs, and outputs a vector 5 that evaluates the ideal square wave given in Eq.1. Your function file header should match the following function S = SquareWave(t.A.T)

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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Qla - Write a function file that takes a domain vector t, a scalar amplitude A, and a scalar period T as inputs, and outputs a vector 5 that evaluates the ideal square wave given in Eq.1. Your function file header should match the following function S = SquareWave(t,A.T) Note: You must write the algorithm yourself. Do not use any built-in functions that explicitly approximate a square waveform. Don't forget to document your function! Q1b Using the function from Q1a, plot a square wave having amplitude A = 1, period T = 2r, over the domain-T ≤ t ≤T. The vector, t, must be resolved to 0.1 units. Use a black dotted line of width 2 when creating your plot. Q1e - Write a function file that takes a domain vector t, a scalar amplitude A, a scalar period T and a scalar representing the number of Fourier series terms n as inputs, and outputs a vector F that evaluates the Fourier series solution given in Eq.2. Your function file header should match the following function F = FourierSqWave(t.A.T.n) Q1d Using the same domain, amplitude, and period as specified in Q1b with the input vector t again resolved to 0.1 units, evaluate the Fourier series for a square wave F(t) for n 11, 5, 10 and 20 terms. Plot each one on a separate 2-by-2 subplot in a new figure window. Q1e Using a while loop and starting with n=1 determine the minimum number of Fourier terms for the average error Emean between solutions S(t) and F(t) to become less than 0.01. For each iteration, also calculate the maximum error Emax and the corresponding location max- You may refer to the appendix for the error calculation formulas. Inside your loop, use fprintf() to output every 10th iteration as a table for tmax Emax and Emean accurate to 4 decimals. For example, the first 3 values of the table should be: t Emax Emax Enean 1 10 20 3.1568 3.1568 3.1568 1.0194 1.1928 1.3797 0.3433 0.0555 0.0382 Your table must include results for n = 1 and the final value at which the 0.01 accuracy is reached. Note: You will need to store each iteration, as elements in vectors for Emean, Emax, and tmax to complete question Q1g.
Transcribed Image Text:Qla - Write a function file that takes a domain vector t, a scalar amplitude A, and a scalar period T as inputs, and outputs a vector 5 that evaluates the ideal square wave given in Eq.1. Your function file header should match the following function S = SquareWave(t,A.T) Note: You must write the algorithm yourself. Do not use any built-in functions that explicitly approximate a square waveform. Don't forget to document your function! Q1b Using the function from Q1a, plot a square wave having amplitude A = 1, period T = 2r, over the domain-T ≤ t ≤T. The vector, t, must be resolved to 0.1 units. Use a black dotted line of width 2 when creating your plot. Q1e - Write a function file that takes a domain vector t, a scalar amplitude A, a scalar period T and a scalar representing the number of Fourier series terms n as inputs, and outputs a vector F that evaluates the Fourier series solution given in Eq.2. Your function file header should match the following function F = FourierSqWave(t.A.T.n) Q1d Using the same domain, amplitude, and period as specified in Q1b with the input vector t again resolved to 0.1 units, evaluate the Fourier series for a square wave F(t) for n 11, 5, 10 and 20 terms. Plot each one on a separate 2-by-2 subplot in a new figure window. Q1e Using a while loop and starting with n=1 determine the minimum number of Fourier terms for the average error Emean between solutions S(t) and F(t) to become less than 0.01. For each iteration, also calculate the maximum error Emax and the corresponding location max- You may refer to the appendix for the error calculation formulas. Inside your loop, use fprintf() to output every 10th iteration as a table for tmax Emax and Emean accurate to 4 decimals. For example, the first 3 values of the table should be: t Emax Emax Enean 1 10 20 3.1568 3.1568 3.1568 1.0194 1.1928 1.3797 0.3433 0.0555 0.0382 Your table must include results for n = 1 and the final value at which the 0.01 accuracy is reached. Note: You will need to store each iteration, as elements in vectors for Emean, Emax, and tmax to complete question Q1g.
BACKGROUND An ideal square wave as pictured in Figure 1.1 is a non-sinusoidal waveform where the amplitude transitions instantaneously between a fixed minimum and maximum periodically. Square waveforms are used in a wide range of applications from electrical switches and clock timers to sound reproduction and image processing. Ampilude Figure 1.1-Square Wave A square wave with amplitude A and period I can be defined by the piecewise function 0<t<T -T<t<0' Eq. 1 A, s(t) = (-A. In practice, an ideal square wave can never be realised as the physical processes that generate the waveform always take a finite amount of time to transition between maximum and minimum values (i.e., they cannot act instantaneously). Therefore, these processes can introduce errors known as artifacts around the step-discontinuity of square waves when t = T. F(t) = A Fourier series consists of an infinite summation of sine waves of different amplitudes, phases, and frequencies and can be used to model any waveform - even non-sinusoidal ones! In practice, we are limited to a finite number of Fourier series terms. The Fourier series for a square wave is given by 11-00 4A T Period k=1 sin((2k-1) wt) 2k-1 1 4A =[sin (wat)+sin(3wt)+sin(5wt) +...], 1 Where w = 2π/T and n is the number of Fourier terms. Eq.2
Transcribed Image Text:BACKGROUND An ideal square wave as pictured in Figure 1.1 is a non-sinusoidal waveform where the amplitude transitions instantaneously between a fixed minimum and maximum periodically. Square waveforms are used in a wide range of applications from electrical switches and clock timers to sound reproduction and image processing. Ampilude Figure 1.1-Square Wave A square wave with amplitude A and period I can be defined by the piecewise function 0<t<T -T<t<0' Eq. 1 A, s(t) = (-A. In practice, an ideal square wave can never be realised as the physical processes that generate the waveform always take a finite amount of time to transition between maximum and minimum values (i.e., they cannot act instantaneously). Therefore, these processes can introduce errors known as artifacts around the step-discontinuity of square waves when t = T. F(t) = A Fourier series consists of an infinite summation of sine waves of different amplitudes, phases, and frequencies and can be used to model any waveform - even non-sinusoidal ones! In practice, we are limited to a finite number of Fourier series terms. The Fourier series for a square wave is given by 11-00 4A T Period k=1 sin((2k-1) wt) 2k-1 1 4A =[sin (wat)+sin(3wt)+sin(5wt) +...], 1 Where w = 2π/T and n is the number of Fourier terms. Eq.2
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