HW 11.5 Consider the periodic "square wave" signal defined by x(t)- u(t - 4k) - u(t - 2-4k) (a) S...
Problem (3) a) A periodic square wave signal x(t) is shown below, it is required to answer the below questions: x(t) 1. What is the period and the duration of such a signal? 2. Determine the fundamental frequency. 3. Calculate the Trigonometric Fourier Series and sketch the amplitude spectrum and phase spectrum of the signal x(t) for the first 5 harmonics. b) Find the Continuous Time Fourier Series (CTFS) and Continuous Time Fourier Transform (CTFT) of the following periodic signals...
A square wave of amplitude A and period T can be defined as -A, 5<t<0, with f(t) = f(t + T), since the function is periodic. Compute the Fourier series for the function in the form f(t) = aneinwot, n=- where wo = 21/T and the coefficients an are the complex Fourier coefficients. Show all your work. Make a simple sketch of the signal and its series. The FIR filter is defined by the filter coefficients bk = [3,-1,2,1] Write...
in MATLAB plot the following EXAMPLE 4.2 Fourier series of a square wave Consider the square wave of Figure 4.4. This signal is common in physical systems. For ex- ample, this signal appears in many electronic oscillators as an intermediate step in the gener ation of a sinusoid We now calculate the Fourier coefficients of the square wave. Because V, 0< t < To/2 x(t) = from (4.23), it follows that ToJTo2 To/2 - e ikast To/2 The values at...
Problem 2: Consider the following periodic signals x(t), a square wave, and yt), a saw tooth 2T The pulses width of x(t) т, wave. Both have the same amplitude A and the same frequency - equal T. The duty-cycle of x(t) is defined as d- T. -A From tables of Fourier Series ofvarious periodic signals, the following formulas are given for your convenience x(= Ad+2Adnacos at+2Ad sna cos 2at+2Adl sun 3xdcos3at яd 2лd Зяd 24 (sin a 1 sin 2asin3ajain...
8.3 Consider the periodic wave equation for t E R and x e T. Suppose the initial conditions are or (0, x) h(x), for g є Cm +1 (T) and h є Cm(T), for m E N. 152 8 Fourier Series (a) Assuming that u(t, x) can be represented as a Fourier series u(t, x)- av(l)ekx, kEZ (8.48) find an expression for ak() in terms of the Fourier coefficients of g and h.
(b) Let X(ju) denote the Fourier transform of the signal r(t) shown in the figure x(t) 2 -2 1 2 Using the properties of the Fourier transform (and without explicitly evaluating X(jw)), ii. (5 pts) Find2X(jw)dw. Hint: Apply the definition of the inverse Fourier transform formula, and you can also recall the time shift property for Fourier Transform. (c) (5 pts) Fourier Series. Consider the periodic signal r(t) below: 1 x(t) 1 -2 ·1/4 Transform r(t) into its Fourier Series...
4. (20 pts) Consider the N-periodic square wave given by 1, 1 sns1 (a) (10 pts) What is the fundamental frequency of the signal an wheN- (b) (10 pts) Find the Fourier series coefficients X[o] and X of the signal xln when N 4. (20 pts) Consider the N-periodic square wave given by 1, 1 sns1 (a) (10 pts) What is the fundamental frequency of the signal an wheN- (b) (10 pts) Find the Fourier series coefficients X[o] and X...
Signal system 8. Consider the periodic signal x(t) = cos(2nt) + cos(2nt) I. a. Find the Fourier series coefficients for this signal. (4 points). b. If this signal passes through a LTI system with the impulse response h(t)=e* u(t), particularly, would the output signal also be a periodic signal ? If so, what would be the Fourier coefficients of the output signal ? (4 points). c. Give the mathematical expression for the output signal y(t). (2 points).
Let x(t) = t, 0<t 1 and Fourier series coefficients a , be a periodic signal with fundamental period of T 2 -t,-1t0 dz(t) a) Sketch the waveform of r(t)d3 marks) b) Calculate ao (3 marks) c) Determine the Fourier series representation of gt)(4 rks) d) Using the results from Part (c) and the property of continuous-time Fourier series to dr(t) determine the Fourier series coefficients of r(t) (4 marks)
l, f) is a periodic signal with period f(t)-n(t)-u(t-t/2 ) for 0 2π a.) Find the exponential Fourier series of f() and sketchf). What is the fundamental radian frequency. b.) Evaluate and sketch |Dml, the magnitude of the Fourier series coefficients vs.o in the range of -4s n S4 c.) Evaluate and sketch the phase angle of D, vs. co in the same range (-4S n S4) d.) Find the signal average power e) Find the approximate average power of...