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N 1 N E[x_kx_k-n] = \sum_{k=1}^N \sin \frac{2\pi k}{N} \sin \frac{2\pi (k - n)}{N} = 0.5 \cos \frac{2\pi n}{N}; \qquad n = 0, 1 \qquad (2.20) E[d_kx_k-n] = \frac{2}{N} \sum_{k=1}^N \cos \frac{2\pi k}{N} \sin \frac{2\pi (k - n)}{N} = -\sin \frac{2\pi n}{N}; \qquad n = 0, 1 \qquad (2.21) Suppose that each waveform is sampled at t = 0, T, 2T,..., such that there are exactly N samples per cycle in the first two cases, and N/2 samples per cycle in the third case. If a sample is taken where f(t) is discontinuous, give it the value just to the right of the discontinuity. Find E[u_ku_k-n]. Find E[u_kx_k-n].

          N
1
N
E[x_kx_k-n] = \sum_{k=1}^N \sin \frac{2\pi k}{N} \sin \frac{2\pi (k - n)}{N}
= 0.5 \cos \frac{2\pi n}{N}; \qquad n = 0, 1 \qquad (2.20)
E[d_kx_k-n] = \frac{2}{N} \sum_{k=1}^N \cos \frac{2\pi k}{N} \sin \frac{2\pi (k - n)}{N}
= -\sin \frac{2\pi n}{N}; \qquad n = 0, 1 \qquad (2.21)
Suppose that each waveform is sampled at t = 0, T, 2T,..., such that there are exactly N
samples per cycle in the first two cases, and N/2 samples per cycle in the third case. If a
sample is taken where f(t) is discontinuous, give it the value just to the right of the
discontinuity.
Find E[u_ku_k-n].
Find E[u_kx_k-n].

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N
1
N
E[xkxk-n] = ∑k=1^N sin(2πk)/(N) sin(2π(k - n))/(N)
= 0.5 cos(2πn)/(N); n = 0, 1 (2.20)
E[dkxk-n] = (2)/(N) ∑k=1^N cos(2πk)/(N) sin(2π(k - n))/(N)
= -sin(2πn)/(N); n = 0, 1 (2.21)
Suppose that each waveform is sampled at t = 0, T, 2T,..., such that there are exactly N
samples per cycle in the first two cases, and N/2 samples per cycle in the third case. If a
sample is taken where f(t) is discontinuous, give it the value just to the right of the
discontinuity.
Find E[ukuk-n].
Find E[ukxk-n].

Added by James C.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Vipin Kumawat

Step 1

From the given information, we know that u(t) is a square wave and x(t) is a sine wave. We also know that N=4, which means there are 4 samples per cycle for u(t) and x(t). To find E[uu-n], we need to calculate the expectation of the product of two samples of u(t) Show more…

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Find the expectation for the following using the example below 2.20 and 2.21. N=4. u(t) is a square wave and x(t) is a sine wave. N 1 2 Tk 2T(k-n) sin sin N N N k=1 E[xxk-n] 2TN = 0.5 cos n=0,1 N N 2 2 k 2T(k-n) E[dxk-n] cos sin N N N k=1 (2.20) 2TN sin N n=0,1 (2.21) Suppose that each waveform is sampled at t = 0, T, 2T,..., such that there are exactly N samples per cycle in the first two cases, and N/2 samples per cycle in the third case. If a sample is taken where f(t) is discontinuous, give it the value just to the right of the discontinuity. Find E[uu-n]. Find E[ux-n].

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