How to draw a Amplitude and Phase spectrum

[rock42]

The problem statement

Sketch the Amp spectrum of the following...
Additionally, for x3, sketch the phase spectrum...

( j is the imaginary number)

x1(t) = cos(10pi*t) + cos(3pi*t)

x2(t) = cos(10pi*t) + cos(5pi*t) - j*sin(10pi*t)

x3(t) = cos(10pi*t + (pi/6)) + j*sin(10pi*t)

The attempt at a solution

For x1 I am fairly certain that the plot should appear as two lines at f = 3/2 and f = 5 with A = 1 for both. I am only confident in this answer as the signal is real and not complex.

For x2 I have drawn two lines once more for f = 1/5 and f = 2/5 with respective A values of sqrt(2) and 1

For x3 I have f = 1/5 and a value of sqrt(2) for it. I have no idea how to interpret the phase spectrum.

There was considerably more to the problem, but I have completed everything but these plots, which I simply cannot find how to draw online. I found one source, but was weary as it did not explain what to do in the event of a phase shift. Does a phase shift effect the Amp spectrum?

 

[rude man]

Don't know what an "Amp spectrum" is, nor a "phase spectrum". Is this in relation to Fourier transform?

 

[rock42]

We have yet to be formally taught the Fourier transform, but an Amplitude spectrum is a plot of amplitude vs frequency and a phase spectrum is a plot of phase shift vs frequency.

 

[rude man]

The problem statement

Sketch the Amp spectrum of the following...
Additionally, for x3, sketch the phase spectrum...

( j is the imaginary number)

x1(t) = cos(10pi*t) + cos(3pi*t)

x2(t) = cos(10pi*t) + cos(5pi*t) - j*sin(10pi*t)

x3(t) = cos(10pi*t + (pi/6)) + j*sin(10pi*t)

The attempt at a solution

For x1 I am fairly certain that the plot should appear as two lines at f = 3/2 and f = 5 with A = 1 for both. I am only confident in this answer as the signal is real and not complex.

That's right. Amplitude spectra convey no information as to phase,only amplitude.

For x2 I have drawn two lines once more for f = 1/5 and f = 2/5 with respective A values of sqrt(2) and 1

Better check your f's on that one! For the rest, see below.

For x3 I have f = 1/5 and a value of sqrt(2) for it. I have no idea how to interpret the phase spectrum.

Again, what is the frequency in Hz? :uhh:

Your amplitude is (probably) not correct. But I don't know how to find it!

The thing is, the problem's terminology is not only unconventional but downright misleading. It mixes phasors and time functions. Complex representations of time functions are phasors and are not time-dependent. Neither are they frequency-dependent. A general phasor is Ae^jθ where A is the phasor amplitude and θ is its phase angle.

So the expression "jsin(10πt)" is basically nonsense, and I don't know what to do with it really.

You could maybe interpret "cos(10πt + π/6)" as as phasor with amplitude 1/√2 and phase angle π/2 + π/6. Why? Because sin(10πt) can arbitrarily be defined to have zero phase. That transforms sin(10πt) to a phasor of 1/√2, jsin(10πt) to a phasor of j(1/√2),
10cos(10πt) to a phasor of (1/√2)e^jπ/2 and cos(10πt + π/6) to (1/√2)e^{j(π/2 + π/6)}. So the total phasor of x3 would be X3 = (1/√2){e^{j(π/2 + π/6)} + j}. But that's just a WAG until they supply you with a legitimate time or phasor representation of x3.

NOTE: The 1/√2 is just a definition of phasor amplitude. It's there for a good reason of course. You should know what it is, or find out.

 

[rezeew]

I would first clarify any confusion or doubts about the problem statement. I would ask for any additional information or context that may be relevant in order to accurately draw the amplitude and phase spectrum for each signal.

Assuming that the signals are continuous and periodic, the amplitude spectrum can be plotted using the Fourier transform. The amplitude spectrum will show the magnitude of the different frequency components present in the signal. In this case, for x1, the amplitude spectrum will have peaks at 3/2 and 5 with amplitudes of 1. For x2, there will be peaks at 1/5 and 2/5 with amplitudes of sqrt(2) and 1, respectively.

For x3, a phase shift of (pi/6) means that the signal has been shifted to the left by (pi/6) radians. This will not affect the amplitude spectrum, as it only shows the magnitude of the frequency components. However, it will affect the phase spectrum, which shows the phase difference between the different frequency components. In this case, the phase spectrum will show a phase shift of (pi/6) at the frequency component of 1/5.

In order to plot the phase spectrum, the Fourier transform of the signal should be taken and the phase angle of each frequency component should be calculated. This can be done using the inverse tangent function and plotting the phase angle against the frequency component. For x3, the phase spectrum will show a phase shift of (pi/6) at the frequency component of 1/5.

In summary, the amplitude spectrum shows the magnitude of the different frequency components present in a signal, while the phase spectrum shows the phase difference between these components. A phase shift in the signal will only affect the phase spectrum, not the amplitude spectrum. It is important to clarify any doubts and gather all necessary information before attempting to draw the amplitude and phase spectrum for a signal.