1)A sine wave of frequency (48/pi) Hz is sampled at 120 samples/second and we wish to represent it in exponential form. The resulting representation is given by

a) e^jo.s* pi* n + e^-jo.s*pi*n/2j

b) e^-jo.s* pi* n + e^-jo.s*pi*n /2j

c) e^jo.8 n + e^-jo.s*pi*n/2j

d) e^jo.sn-^-jo.sn/2

2) Find the resolution of the Fourier transform if a signal is sampled at 512 samples/second and we collect a total of 16,384 data points and then apply the FFT algorithm to them.

a) 256

b ) 0.0313

c) 8.192

d)32

3) You are sampling a signal which consists of a mixture of 160, 164 and 168 Hz sinusoids and which you want to resolve in your FFT. If you are storing 4096 samples, what is the range of sampling frequencies you can use such that you don't violate Nyquist theorem and still meet or exceed the needed resolution when you perform the FFT operation on the full set of collected data

a) any frequency greater than or equal to 336 Hz

b)less than or equal to 16384 Hz but greater than 336 Hz

c) less than or equal to 4096 but greater than 336 Hz

d) less than or equal to 8192 Hz but greater than 336

4) given that
\(e^{j\pi } equals -1\)
find the third root of -1 using Euler's formula. In other words, find the value of (-1)^1/3

a) 0.866 (-+) j 0.866

b) 0.5(-+)j0.5

c) 0.5(-+) j 0.866

d) 0.866 (-+) j0.866

5) Using the fundamental definition of discrete Fourier transform (DFT)

X[k] =
\(\sum_{n=0}^{N-1} x(n)e^{-j2\pi kn/N}\)
find the numerical values of the first and last DFT terms (X[0] and X[3] of a periodic sequence with a digital period of 4 if the first four terms of the sequence are given by 6, 4, 4, and 4.

a) X(0) = -18,X(3) = -2+j1

b) X(0) = 18,X(3) = 2+j0

c) X(0) = 18,X(3) = -2+j1

d) X(0) = 18,X(3) = -18+j18