EMJ22403 SIGNAL AND SYSTEMS SEM II 2021/2022 ASSIGNMENT II Name :... Matric:... Group:.. 20 Question The form of the FS

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Emj22403 Signal And Systems Sem Ii 2021 2022 Assignment Ii Name Matric Group 20 Question The Form Of The Fs 1 (100.57 KiB) Viewed 49 times

EMJ22403 SIGNAL AND SYSTEMS SEM II 2021/2022 ASSIGNMENT II Name :... Matric:... Group:.. 20 Question The form of the FS representation presented in this question, namely 00 x(t) = 2x[k]eikwot = k=-00 is termed the exponential FS. this problem, you need to explore several alternative, yet equivalent, ways of expressing the FS representation for real-valued periodic signals. (a) Trigonometric form. i. Show that the FS for a real-valued signal x(t) can be written as 00 x(t) = B[0] + B[k] cos(kw.t) + A[k] sin(kw,t) k=1 Where B[k] and A[k] are real-valued coefficients. (3 marks) ii. Express X[k] in terms of B[k] and A[k]. (3 marks) iii. Use the orthogonality of harmonically related sines and cosines to show that B[0] = x(t)dt, - . - Srce 2 B[k] = x(t) cos kw,tdt And A[k] = 1=" S x(t) sin kw, tdt. (4 marks) iv. Show that A[k] = 0 if x(t) is even and B[k] = 0 if x(t) is odd. (1 marks) EMJ22403 SIGNAL AND SYSTEMS SEM II 2021/2022 ASSIGNMENT 11 (b) Polar form. i. Show that the FS for a real-valued signal x(t) can be written as x(t) = C[0] + C[k] cos(kw,t + e[k]) k=1 Where C[k] is the (positive) magnitude and @[k] is the phase of kth harmonic. (4 marks) Express C[k] and @[k] as a function of X[k]. ii.