Fourier Series 3 3. Someexamples The easiest example would be to set f(t) = sin(2…t). Without even performing thecalculation (simplyinspectequation2.1)weknowthattheFouriertransform We need a fixed time period for those signals. We will take it from the user. We need to find the time domain values. We will find it from the fraction below. Now, derive the message signal as below. Make a subpolt as 3 by 1 size. Place the message signal (upcoming) to first subplot. Set the title. Now, derive the carrier signal as below. we see that the resulting plot resembles the original square wave, as seen in Fig. 3. Fig. 3. Square wave approximation. If the Fourier series were summed from 0 to infinity (an infinite number of terms), the result would be an exact square wave. Using the Fourier Series GUI, try adjusting the frequencies and magnitudes of three sine waves to let the 1st time period have a Duty Cycle 55 % from T = 0 to T=0.1s 2nd time period have a duty cycle 58 % from T = 0.1s to 0.2s 3rd one with duty cycle 56 % from T = 0.2s to 0.3s 4th one with duty cycle 52% from T = 0.3s to 0.4s Now to compensate for the above signal i.e , until T =0.4s I need to generate a signal with parameters as below The computer language: Matlab We have chosen to use the programming lan-guage Matlab, because this language gives very compact and readable code that closely resembles the mathematical recipe for solving the problem at hand. Mat-lab also has a gentle learning curve. There is a Python companion of this book in case that language is preferred. The fft function in MATLAB® uses a fast Fourier transform algorithm to compute the Fourier transform of data. Consider a sinusoidal signal x that is a function of time t with frequency components of 15 Hz and 20 Hz. Use a time vector sampled in increments of 1/50 seconds over a period of 10 seconds. f (x) is single valued, piecewise monotonic and piecewise continuous. Syntax of Fourier Series in Matlab 1. First, we will compute the sine and cos coefficients of Fourier series and also the partial sum of Fourier series. For an expression 'f' we can compute 'nth' sum in the range / interval [-P, P]. Syntax: syms z n P x To represent this in MATLAB, type the following into the MATLAB command window: polynomial = s^4 + 3*s^3 - 15*s^2 - 2*s + 9 polynomial = s^4 + 3 s^3 - 15 s^2 - 2 s + 9 Continuous-time transfer function. Instead of using the roots function, we can use the zero function to find the roots of the polynomial. zero (polynomial) The sawtooth wave is defined to be -1 at multiples of 2 π and to increase linearly with time with a slope of 1/ π at all other times. example x = sawtooth (t,xmax) generates a modified triangle wave with the maximum location at each period controlled by xmax. Set xmax to 0.5 to generate a standard triangle wave. Examples collapse all x = square (t) generates a square wave with period 2 π for the elements of the time array t. square is similar to the sine function but creates a square wave with values of -1 and 1. example x = square (t,duty) generates a square wave with specified duty cycle duty. To do this change one of the signals to a square wave of amplitude = 1 v. Then adjust the truncation window length for as short as possible and then use the calculation point slider to compute the convolution function. The resulting convolution is the same as the other signal. UNIT IMPULSE RESPONSE Here's how to do it in MATLAB