Sketching the Spectrum of the Signal x(t)

[freezer]

Homework Statement

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

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

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

Homework Equations

The Attempt at a Solution

$a_{k} = \frac{1}{1 + j\pi k}$

$a_{-k}^{*} = a_k$

$a_1 = \frac{1}{1+j\pi}$

$a_2 = \frac{1}{1+j2\pi}$

$a_3 = \frac{1}{1+j3\pi}$

Convert them to polar, I get the general form:

$\frac{1}{\sqrt{(k\pi)^2 + 1}}e^{-j arctan(k\pi)}$


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.

 

[rude man]

This all looks good. The frequencies are +/- 0, 2, 4 and 6 Hz as you post.
Did you remember a<sub>0/SUB] = 1?

If all you're supposed to do is plot the magnitude spectrum, you'd be done.

The exp(-jtan^-1kπ) term is the phase expression. I.e. phase = tan^-1(kπ) and you could plot that vs. the 7 frequency spots as well.</sub>

 

[freezer]

Thank you for checking my work. I was concerned about the arctan, i had not come across that before.

the other way i was thinking if we look at k=+/-1

$\frac{1}{1+j\pi}e^{j\pi t} + \frac{1}{1-j\pi}e^{-j\pi t}$

Getting a common denominator

$\frac{2}{1+ \pi^2}e^{j\pi t} + \frac{2}{1+\pi^2}e^{-j\pi t}$

for k=+/-2

$\frac{2}{1+4 \pi^2}e^{j4\pi t} + \frac{2}{1+4\pi^2}e^{-j4\pi t}$

and for k = +/-3

$\frac{2}{1+9 \pi^2}e^{j12\pi t} + \frac{2}{1+9\pi^2}e^{-j12\pi t}$

and yes, i did forget to mention k=0 if 0 with mag of 1.

 

[rude man]

Thank you for checking my work. I was concerned about the arctan, i had not come across that before.

You will, many times I'm sure.

Reason: a + jb = √(a² + b²) exp[ tan^-1(b/a)]
in other words, this is how you change from cartesian to polar & back.

the other way i was thinking if we look at k=+/-1

$\frac{1}{1+j\pi}e^{j\pi t} + \frac{1}{1-j\pi}e^{-j\pi t}$

Getting a common denominator

$\frac{2}{1+ \pi^2}e^{j\pi t} + \frac{2}{1+\pi^2}e^{-j\pi t}$

The second line does not follow from the first. Following what I wrote above,

1/(1 + jπk) = 1/√[1 + (πk)²] exp[j tan^-1(-πk)]

so 1/(1 + jπk) exp(j4πkt) = 1/√[1 + (πk)²] exp[j tan^-1(-πk)] exp(j4πkt)

= 1/√[1 + (πk)²] exp j[4πkt + tan^-1(-πk)].

When you do any arc tan function, remember arc tan(-b/a) is not the same angle as arc tan(b/-a)
so don't leave the expression as exp[-j arc tan(πk)] as you did in your post #1. It should be exp[j arc tan(-πk)].

As I said, arc tan(-πk) is the phase angle, and arc tan(-πk/1) and arc tan(πk/-1) are 180 degrees apart!

 

[rude man]

You will, many times I'm sure.

CORRECTION:
Reason: a + jb = √(a² + b²) exp[j tan^-1(b/a)]
in other words, this is how you change from cartesian to polar & back.


The second line does not follow from the first. Following what I wrote above,

1/(1 + jπk) = 1/√[1 + (πk)²] exp[j tan^-1(-πk)]

so 1/(1 + jπk) exp(j4πkt) = 1/√[1 + (πk)²] exp[j tan^-1(-πk)] exp(j4πkt)

= 1/√[1 + (πk)²] exp j[4πkt + tan^-1(-πk)].

When you do any arc tan function, remember arc tan(-b/a) is not the same angle as arc tan(b/-a)
so don't leave the expression as exp[-j arc tan(πk)] as you did in your post #1. It should be exp[j arc tan(-πk)].

As I said, arc tan(-πk) is the phase angle, and arc tan(-πk/1) and arc tan(πk/-1) are 180 degrees apart!

See correction above.