- #1

- 76

- 0

## Homework Statement

Make a sketch of the spectrum of the signal defined by:

[itex] x(t) = \sum_{k = -3}^{3}\frac{1}{1+j\pi k}e^{j4\pi kt} [/itex]

Use polar notation for the phasors on the plot, and sketch the frequency axis in Hz.

## Homework Equations

## The Attempt at a Solution

[itex]a_{k} = \frac{1}{1 + j\pi k}[/itex]

[itex]a_{-k}^{*} = a_k[/itex]

[itex]a_1 = \frac{1}{1+j\pi} [/itex]

[itex]a_2 = \frac{1}{1+j2\pi} [/itex]

[itex]a_3 = \frac{1}{1+j3\pi} [/itex]

Convert them to polar, I get the general form:

[itex] \frac{1}{\sqrt{(k\pi)^2 + 1}}e^{-j arctan(k\pi)} [/itex]

My frequency stems will be at -6, -4, -2, 0, 2, 4, 6 with the above amplitude for each of the corresponding k values.

Can you double check my work, the arctan is makeing me question my results.