Linking Fourier Transform, Vectors and Complex Numbers

The Attempt at a Solution

I tried to attempt the question but I am not sure how to start it, at least for part (i).

My biggest question, I think, is how does the multiplication of a random complex number to a Fourier-Transformed signal (V(f)) have an effect on V(f)? Although the question did not specify the type of signal that has been Fourier-Transformed, I shall assume that the original signal is a cosine waveform, as shown below:

with fm=1.

In the question, it also mentioned to visualise V(f) and cV(f) as vectors in the complex plane, but how should I do it? Are there any diagrams online that show me how to represent cV(f) and V(f) as vectors in the complex plane?

Thank you. :)

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I'd suggest you start by answering the following:
1. For some frequency $f$, what does the modulus and argument of $V(f)$ represent? The first paragraph of the Wiki should be helpful.
2. If you have two complex numbers in polar form $z_1 = r_1\angle\phi_1,z_2 = r_2\angle\phi_2$, what is their product $z_1 z_2$ equal to?

I'd suggest you start by answering the following:
1. For some frequency $f$, what does the modulus and argument of $V(f)$ represent? The first paragraph of the Wiki should be helpful.
2. If you have two complex numbers in polar form $z_1 = r_1\angle\phi_1,z_2 = r_2\angle\phi_2$, what is their product $z_1 z_2$ equal to?
1. The modulus of $V(f)$ represents the amplitude of $v(t)$, while the argument of $V(f)$ represents the phase angle of $v(f)$.

2. The product should be equals to $r_1r_2\angle(\phi_1+\phi_2)$. Am I right to say this?

Thank you. :)

1. The modulus of $V(f)$ represents the amplitude of $v(t)$, while the argument of $V(f)$ represents the phase angle of $v(f)$.
The modulus and argument of $V(f)$ represents the amplitude and phase, respectively, of the frequency component at $f$, but you get the idea.

2. The product should be equals to $r_1r_2\angle(\phi_1+\phi_2)$. Am I right to say this?
Yes, you are. For your first question (i), you again have two complex numbers $c$ and $V(f)$, so what can you say about their product $cV(f)$?

Yes, you are. For your first question (i), you again have two complex numbers $c$ and $V(f)$, so what can you say about their product $cV(f)$?
There will be an increase in the amplitude of the wave which has been Fourier Transformed, while for the angle, it would be shifted x radians away to the left, since the angles will add up when $c$ is multiplied to $V(f)$. Am I right to say this? :)

There will be an increase in the amplitude of the wave which has been Fourier Transformed, while for the angle, it would be shifted x radians away to the left, since the angles will add up when $c$ is multiplied to $V(f)$. Am I right to say this? :)
You're talking about the vector interpretation of complex numbers? Like, if $\arg(c)$ is positive, the vector $V(f)$ would be scaled and rotated counterclockwise in the complex plane? That's how I understand what you wrote, and it's true, so it seems you already know everything you need to solve your assignment.

Do you have any questions?

I was visualising it from a mathematical viewpoint, as in the earlier question, multiplying 2 complex numbers, $z_1$ and $z_2$.

Right now, I still can't really link the vector diagram of complex numbers to the plot of amplitude-frequency of waves, I think. >< The latter, from my point of view, is a diagram of spikes coming out at different frequencies, depending on the original wave.

I do agree that the vector $V(f)$ will be rotated anti-clockwise as the summing of angles is positive.

Thanks a lot for your help! :D

Right now, I still can't really link the vector diagram of complex numbers to the plot of amplitude-frequency of waves, I think. >< The latter, from my point of view, is a diagram of spikes coming out at different frequencies, depending on the original wave.
Ah, you mean the amplitude spectrum, e.g:

But that's just a plot of $|V(f)|$ over some interval of $f$, i.e. it's a plot of the modulus of $V(f)$. In some contexts, you also include the phase spectrum, which is just a plot of $\arg(V(f))$.

Does that help?

Ah, you mean the amplitude spectrum, e.g:

But that's just a plot of $|V(f)|$ over some interval of $f$, i.e. it's a plot of the modulus of $V(f)$. In some contexts, you also include the phase spectrum, which is just a plot of $\arg(V(f))$.

Does that help?
I see. Yupp, I meant that. Haha. That was the diagram shown in my lecture notes, so I can link to it better. :)

Yes, and thank you so much for your help! :D I should be fine with my assignment now.