Magnitude and phase of the Fourier transform

In summary, the conversation discusses the Fourier transform of a rectangular pulse and the properties of the transform. The speaker also mentions their belief that a summation or integral of the magnitude of the transform with a cosine term would produce the original function, but is corrected by the other person. They also mention their experience as an electrical engineer and how they usually deal with complex valued signals.
  • #1
PainterGuy
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TL;DR Summary
Trying to understand how a function f(t) could be generated using magnitude and phase information from its FT at basic level.
Hi,

A rectangular pulse having unit height and lasts from -T/2 to T/2. "T" is pulse width. Let's assume T=2π.

The following is Fourier transform of the above mentioned pulse.
F(ω)=2sin{(ωT)/2}/ω ; since T=2π ; therefore F(ω)=2sin(ωπ)/ω

Magnitude of F(ω)=|F(ω)|=√[{2sin(ωπ)/ω}^2]=|2sin(ωπ)/ω|

Phase of F(ω), ∠F(ω): phase of complex number x+iy is defined as: θ=tan⁻¹(y/x). In case of F(ω) "y" is zero. The expression "2sin(ωπ)/ω" would alternate between "+" and "-" values. θ=lim_(y→0){tan⁻¹(y/x)}=0, π. The phase, ∠F(ω), switches between "0" and "π" depending upon the sign of "x".

I have always thought that summation or integral of:
|F(ω)|cos{ωt+∠F(ω)} for ω ≥ 0
would produce the original function. Am I thinking correctly? Thank you!
 
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  • #2
The Fourier transform requires all frequencies I believe.
Even for a real f(t) the transform F(ω) is not necessarily real. A sufficient condition is that that if f(t)=f(-t) then F(ω) is real. But any translation in time by t0 will give e-iωt0F(ω).
 
  • #3
I'm sorry but it looks like you're missing my question. Anyway, yes, FT requires all frequencies. Trigonometric form uses only positive frequencies while exponential form of FT used negative and positive frequencies to make things symmetric.
 
  • #4
PainterGuy said:
I have always thought that summation or integral of:
|F(ω)|cos{ωt+∠F(ω)} for ω ≥ 0
would produce the original function. Am I thinking correctly? Thank you!
No.

When you have such questions, the best thing to do is start with the definition and just do the algebra:
$$ \begin{eqnarray*}
f(t) & = & \frac{1}{2\pi}\int_{-\infty}^\infty F(\omega) \, e^{i\omega t} \, d\omega \\
& = & \frac{1}{2\pi}\int_{-\infty}^\infty \left| F(\omega)\right| \, e^{i \angle F(\omega)} \, e^{i\omega t} \, d\omega \\
& = & \frac{1}{2\pi}\int_{-\infty}^\infty \left| F(\omega)\right| \left[ \cos\left(\omega t + \angle F(\omega) \right) + i \sin \left(\omega t + \angle F(\omega) \right)\right]
\end{eqnarray*}
$$

Now, if your ##f(t)## has special properties, like being real, then there are symmetries that you can use to modify this expression and perhaps throw out terms, but usually that doesn't actually help you compute anything . As an electrical engineer I usually deal with signals that are complex valued, so I always use the full definition of the inverse transform.

Note: I edited this post significantly a few minutes after the initial posting.

jason
 
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  • #5
jasonRF said:
As an electrical engineer I usually deal with signals that are complex valued, so I always use the full definition of the inverse transform.

Thank you for the help. Now I can see where I was going wrong. I should have used complex sinusoid.
 

1. What is the Fourier transform and why is it important?

The Fourier transform is a mathematical operation that decomposes a signal into its frequency components. It is important because it allows us to analyze and manipulate signals in the frequency domain, which can provide valuable insights and applications in various fields such as signal processing, image processing, and data compression.

2. What is the difference between magnitude and phase in the Fourier transform?

The magnitude of the Fourier transform represents the amplitude or strength of each frequency component in the signal, while the phase represents the relative timing or position of each component. In other words, the magnitude tells us how much of each frequency is present in the signal, while the phase tells us when each frequency occurs.

3. How is the magnitude and phase of the Fourier transform calculated?

The magnitude is calculated by taking the absolute value of the complex numbers obtained from the Fourier transform. The phase is calculated by taking the arctan of the imaginary part divided by the real part of the complex numbers. Both the magnitude and phase can also be visualized using a frequency spectrum plot.

4. What is the significance of the magnitude and phase in signal analysis?

The magnitude and phase provide important information about the characteristics of a signal. The magnitude can help identify the dominant frequencies in a signal, while the phase can reveal the relative timing and relationships between different frequency components. This information can be used for various applications such as filtering, noise reduction, and feature extraction.

5. Can the magnitude and phase of the Fourier transform be altered?

Yes, the magnitude and phase of the Fourier transform can be altered through various methods such as filtering, modulation, and phase shifting. These techniques can be used to manipulate the frequency components of a signal and achieve desired effects such as noise reduction, signal enhancement, and data compression.

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