Guess the Periodic Signal
A certain periodic signal has the following properties:
1. N = 4
2. $ \sum_{n=0}^{3}x[n] = 4 $
3. $ \sum_{n=1}^{4}(-1)^nx[n] = 2 $
4. For even $ k\, $'s, $ a_k = a_{k+1}\, $
Answer
From 1. we know that $ x[n] = \sum_{n=0}^{3}a_k e^{jk\frac{\pi}{2}n}\, $
Using 2., it is apparent that this is the formula for $ a_k\, $. Specifically, for $ a_0\, $, since the only thing under the sum is $ x[n]\, $. So,
Now that we know $ a_0\, $, we know that $ x[n] = 1 + \sum_{n=1}^{4}a_k e^{jk\frac{\pi}{2}n}\, $
Since $ \omega_0 = \frac{\pi}{2}\, $, let's try and find $ a_2\, $,
$ a_2 = \frac{1}{4}\sum_{n=0}^{3}x[n] e^{-2j\frac{\pi}{2}n} = \frac{1}{4}\sum_{n=0}^{3}x[n] e^{-j\pi n}\, $
Using the property that $ e^{-j\pi n} = (e^{-j\pi})^{n} = (-1)^{n} \, $, we can change the above equation to
$ a_2 = \frac{1}{4}\sum_{n=0}^{3}x[n](-1)^{n}\, $
Since the function is periodic and the $ a_k\, $'s repeat every 4 integers, we are able to shift the bounds of summation by one.
According to 3. $ \sum_{n=1}^{4}(-1)^nx[n] = 2 $, and
$ a_2 = \frac{1}{4}\sum_{n=1}^{4}x[n](-1)^{n}\, $
$ a_2 = \frac{1}{4} * 2\, $, and
$ a_2 = \frac{1}{2}\, $
Since we know the 2 even $ a_k\, $'s in the fundamental period, by property 4, we can find the 2 odd $ a_k\, $'s.
$ a_0 = a_1 = 1\, $
$ a_2 = a_3 = \frac{1}{2}\, $
Now, the periodic signal has been found to be
$ x[n] = \sum_{n=0}^{3}a_k e^{jk\frac{\pi}{2}n}\, $
$ x[n] = 1 + \frac{1}{2}e^{j\frac{\pi}{2} n} + e^{j\pi n} + \frac{1}{2}e^{j\frac{3\pi}{2}n}\, $
