Determine the Periodic Signal
A periodic signal x[n] has the following characteristics:
1. N = 4
2. The average value of the signal over the interval $ 0 \leq n \leq 7 $ is 0.
3. $ \sum_{n=0}^{3}x[n](-j)^n = -20j $
4. $ a_{-k} \; = \; a_{k} $
Solution
Coupling 1. and 2. together, we see that the interval is twice the fundamental period. The coefficient $ a_{0} $ is equal to the average value of the signal over one period, so $ a_{0} = \frac{0}{2} = 0 $.
Recall that $ a_{k}=\frac{1}{N} \sum_{n=0}^{N-1} x[n]e^{-jk \frac{2\pi}{N}n} = \frac{1}{4} \sum_{n=0}^{3} x[n]e^{-jk \frac{\pi}{2}n} $
If k = 1, then $ a_{1}=\frac{1}{4} \sum_{n=0}^{3} x[n]e^{-j \frac{\pi}{2}n} $, and, realizing that $ e^{-j \frac{\pi}{2}n} = (-j)^n $, $ a_{1}=\frac{1}{4} \sum_{n=0}^{3} x[n](-j)^n $.
By 3., $ \sum_{n=0}^{3} x[n](-j)^n = -20j $, so $ a_{1}=\frac{1}{4}(20j) = -5j $.
Because N = 4, $ a_{3} = a_{-1} $, and by 4. $ a_{-1} = -a_{1} $, so
$ a_{3} = 5j $ .
