2
votes
Accepted
Relationship between fourier transform and fourier series
Transform looks right, but the logic afterwards needs some correction.
$$
X(\omega) =
-i \pi A \left(e^{i \alpha } \delta \left(\omega -2 \pi f_0\right)-e^{-i \alpha } \delta \left(2 \pi f_0+\...
- 46
2
votes
Accepted
Fourier transform of $|x_\mathrm{a}(t)|^2$
Your result is correct, but it can be simplified by noticing that the two integrals in your solution are actually identical, which for $f>0$ leads to
$$\mathscr{F}\left\{|x_\mathrm{a}(t)|^2\right\}=...
- 88.8k
1
vote
Fourier transform magnitude of the sum of two signals
In general we have
$$\big|X_1(f)+X_2(f)\big|\neq \big|X_1(f)\big|+\big|X_2(f)\big|\tag{1}$$
However, the condition $X_1(f)X_2(f)=0$ $\forall f$ implies that for any $f$, either $X_1(f)$ or $X_2(f)$ or ...
- 88.8k
1
vote
Accepted
Why does applying Fourier Transform on point Spread Function yield h(t) which is complex-valued
The Fourier transform doesn't move only from time to frequency (or space etc.), it moves between these domains. The only difference between the Fourier transform and its inverse is a sign in the ...
- 88.8k
1
vote
Difference between DC component and zero frequency component of signal
A brief answer.
Rect does NOT contain any DC component = No component at frequency = 0.
But the fourier transform does tell there is a DC component at frequency = 0, and the rect function associated ...
1
vote
Accepted
Is it useful to think of a Fourier Transform as writing out a signal in terms of a basis?
Question 1: Yes. In fact, the Fourier series becomes countable, although infinite, under restriction, say $[0,2\pi]$, with some exceptions. See this: Schauder basis: Relation to fourier series.
...
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1
vote
Accepted
Alternative way to find fourier transform
There are two main strategies to simplify the calculation of the Fourier Transform.
Use Fourier Transform properties
"Deconstruct" the time domain functions into other functions with easier ...
- 42.5k
1
vote
Alternative way to find fourier transform
With such a simple function it is probably easiest to directly solve the Fourier integral. It's also wise to commit such simple Fourier identities to memory.
Another relatively simple method I can ...
- 88.8k
1
vote
Why does multiplying a real signal by a random complex phase term result in "spreading" in the Fourier domain?
I'm not sure I correctly understand what you mean by DC term, but will assume you mean fundamental frequency $f_0$ of some periodic signal, i.e. an unmodulated carrier.
So your real-valued signal $f(x)...
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