Does anyone have a mechanism to understand intuitively ( and automatically ) why the fourier transform of certain functions have certain shapes ( at least for some functions, not necessarily for all ) ? I know what kind of operator the fourier transform is and what it does to a function but somehow i can't see intuitively and automatically why why the fourier transform of certain functions have certain shapes. For example , is there a intuitive reason the fourier transform of a pulse ( box function ) is in the sync shape ?
Thanks in advance
Fourier Transform
fouriersignal processing
Related Solutions
You don't say what you mean by a "ramp function" so I'll just show you how pictorially, analytically you can get your function that you need.
remember that multiplication in the time domain is the same as convolving in the frequency domain and the converse is true too. Convolving in the time domain is the same as multiplying in the frequency domain.
I know the relationships between several shapes in both the time and frequency domains. One I've chosen is the rect function (rectangle) which has a sinc(f) frequency function.
I then think about how to generate a ramp function that increases and then decreases.
Here is shown two rect functions one red and one black (for ease of understanding). Figure 1 is before they intercept. 2 is when they start to intercept (with the black rectangle being the over lap area) 3 is at maximal overlap (black area is maximal) 4. is when the overlap is decreasing as the red passes through the black rect and 5. is when it is over.
Figure 6 shows the result of this convolution with a ramp up and then a ramp down.
Now the fun part.
A convolution of two rect functions in the time domain is a multiplication in the frequency domain. Since the two rect's are the same we simply get \$sinc^2\$ in the frequency domain.
I'll leave it to you to fill in the details of the actual mathematic steps. the combining and use of fundamental functions is the key insight you need.
Intuitively the DFT and FT are similar. As the other poster pointed out, the main difference is that sampling in the time domain is equivalent to repetition in the Frequency domain, so the DFT will have repeated spectra at multiples of the sample frequency.
Two problems here.
i) abs(real(y)) is incorrectly being used. You need the absolute value (or magnitude) of both the imaginary and real components ... abs(y).
ii) You're seeing the effect of spectral leakage because the DFT assumes a repetitive signal; i.e. the maths "assumes" the signal you applied is repeated infinitely. Your signal will have discontinuities in this case.
Try the following code:
t = [-10*pi:pi/10:9.9*pi]
x = sin(t)
y = fft(x)
z = abs(y)
You should see this:
Related Topic
- Electronic – Why isn’t the Fourier transform of a single sine wave cycle a single bar
- Electronic – Fourier Transform and the Delta Function
- Electrical – Interpretation of Laplace transformed function and Laplace vs Fourier
- Electronic – fast Fourier Transform difference with discrete Fourier Transform
- Electronic – Real time spectrum analyzer and Fourier Transform
- Electronic – What are the applications of the Fourier transform in communications
Best Answer
I am not sure about intuition in general, but regarding the step-function FT being a sync function:
Note that the shape will remain the same, but the frequencies over which the FT of a particular step-function resides is a function of the pulse-width of the original signal. Namely, expanding a function in the time-domain actually shrinks the corresponding frequency-domain function (think slowing down voice recordings, the sound gets very low i.e. lower frequency).
That being said, as you decrease the pulse-width of a particular step-function the frequency components of that signal increase because now there is more change happening (to use a loose descriptor) in a shorter amount of time.
In contrast, if we expand the step-function in the time-domain to have a longer pulse-width then there is less change and the corresponding frequency components must be much lower.
In general, I look at a function and try and get a feel for how quickly it might be changing to get a rough idea. But as I said, I don't know of any general rule of thumb here.