Introduction to Digital Signal and System Analysis - Tutorsindia

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Introduction to Digital Signal and System Analysis Frequency Domain Analysis 4 Frequency Domain Analysis 4.1 Fourier series for periodic digital signals Consider a periodic digital signal x[n], n=0,1,2,...,N-1, where N is the number of sample values in each period. From Euler’s complex exponential equation: 2π kn 2π kn 2π exp − j = cos − j sin k=0,1,2,...,N-1 (4.1) N N Nkn for each frequency k, Eq. (4.1) contains 2 sinusoidal functions in real and imaginary parts, respectively, with p/2 difference in phase. The frequencies of the functions are k=0,1,2,...,N-1. The fundamental sinusoidal function is when the frequency k=1. The other higher sinusoidal functions are called harmonics. The coefficients of Fourier Series for a digital signal can be calculated by where a − 1 1 N k = N n= 0 x[ n]exp − 2π kn j N k=1,2,…,N-1 (4.2) a k is kth spectral coefficient, indicating the strength of the kth harmonic function. The original digital signal x[n] can be represented by its constituent harmonics as the form of discrete Fourier series: x[ n] = N − 1 k = 0 a k 2π kn exp j . N (4.3) The following Figure 4.1 illustrates how many complex multiplications are required in calculating those coefficients 2 and from a k to obtaining x [n] . It is basically N in each occasion. ak 42 Download free ebooks at bookboon.com
Introduction to Digital Signal and System Analysis Frequency Domain Analysis Figure 4.1 Number of complex multiplications Example 4.1 A periodic signal x [ n] = [0 2 4 6] has the period N=4, its coefficients of Fourier series can be calculated by the formula Do you want your Dream Job Please click the advert More customers get their dream job by using RedStarResume than any other resume service. RedStarResume can help you with your job application and CV. Go to: Redstarresume.com Use code “BOOKBOON” and save up to $15 (enter the discount code in the “Discount Code Box”) 43 Download free ebooks at bookboon.com
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