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Frequency Domain Signals
Fourier theory
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Fourier Domain
Previous section everything was shown in the time domain.
The same operations can be performed in the frequency domain. To do this we need to convert time domain signals into the frequency domain.
We shall use Fourier Transforms to do this.
Being able to operate in either the time domain or frequency domain has advantages both in terms of simplification of operations and compute time (in digital signal processing)
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Fourier Domain
Recall that we can represent signals by the sum of other signals
Here is an example with a square wave
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Fourier Domain
We can find the amount of different signals that are contained in our signal by seeing how similar they are.
Recall we did this by multiplying our signal by another signal and summing the result.
Here we multiply our square wave by cosine waves with different frequencies.
We can see that the result of the sum is zero for even harmonics of the cosine waves and non-zero for the odd harmonics.
What do you think will happen if we use sine waves?
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Fourier Domain
For sine waves all the harmonics, both odd and even, sum to zero.
This is because our square wave is an even function and the sine wave is an odd function.
This means that we can make our square ware with just cosine terms.
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Fourier Domain
The trigonometric Fourier series is:
We can use symmetry to simplify the reconstruction in this case.
For our square wave example, the signal is even so the sine coefficients will be zero as sine is an odd function.
We don’t need many terms to get a decent square wave.
The even orders also have zero coefficients, again due to symmetry.
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Fourier Domain
We can find the value for each coefficient by multiplying our signal by each harmonic and sum the result.
Odd harmonics are not symmetrical above and below the zero line. So will have a non zero value when summed. (see earlier example for more detail)
Even harmonics are symmetrical about the zero line and so they sum to zero and do not contribute to the approximation.
a0 = 0.5 (the mean of our square wave)
a1 = 0.63, a2=0, a3=-0.21, a4=0 …
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Fourier Domain
For Odd functions we’d use sines
If the function was neither odd or even then we need both terms.
To get an approximation of the signal we need to solve three integrals / summations.
Integrate to get average
Integrate for cosine terms
Integrate for sine terms
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Fourier Domain
Use one integral instead? Exponential Fourier Series
We know Euler’s formula links e and sin & cos so we can use this to link the two versions, if they both represent the same signal they should be equivalent.
For n=0, a0 = co, both are the average signal
an = 2 real(cn)
bn=-2 imag(cn)
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Fourier Domain
The Fourier Transform of a function can be derived as a special case of the Fourier Series when the period, T→∞,
Fourier Transform pair is:
If t is measured in seconds then f is in cycles per second or Hz.
If t is measured in metres, then H is a function of spatial frequency and f is in cycles per meter.
The Fourier transform is a linear operator, the transform of two summed functions is the sum of the transforms of the two functions.
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Fourier Domain
For Fourier transforms on discrete signals we use the Discrete Fourier Transform (DFT).
Practically this is usually via the Fast Fourier Transform (FFT) though I will use the terms interchangeably, (remember that the FFT has some specific requirements for the signal)
*You may often see a 1/N scaling factor in the inverse
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Fourier Theorems
Fourier scaling theorem
If you scale the time axis by a this changes the frequency scale and the amplitude of the response by 1/a
Example: t = 2t
We see the amplitude of Fourier component scaled by ½
Component at the same frequency but points twice as close together
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Fourier Theorems
Fourier shift theorem
This states that a delay in the time domain corresponds to a linear phase term in the frequency domain.
A delay of Δ gives a linear phase term of multipled by the unshifted spectrum.
This means that you are looking at the magnitude of the spectrum it will be unchanged, only the phase is modified.
Shift was 0.25 seconds
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Fourier Theorems
Fourier convolution theory
This is one of the most useful theorems as the consequences allow the simplification or speed up of many calculations.
This is the DFT of Y but shifted by m, so using the shift theory we get:
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Fourier Theorems
This is the DFT of x
So convolution of two signals in time is equal to the multiplication of their spectra
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Fourier Theorems
This is so useful because a FFT and multiply and a corresponding IFFT is often much faster to compute than direct calculation of the convolution.
This is because convolution scales with N2 where as the operations for FFT scale Nlog(N)
So we have (N+2Nlog(N)) for FFT based convolution
Actual compute speeds may be different due to memory usage, setup overheads, other optimisations etc.
N NxN N+2Nlog(N) ratio
4 16 20 0.8
8 64 56 1.1
16 256 144 1.7
32 1024 352 2.9
64 4096 832 4.9
128 16384 1920 8.5
256 65536 4350 15
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Fourier Theorems
Also this applies the other way around.
A multiply in the time domain is equivalent to a convolution in the frequency domain.
It also explains why windowing a function in the time domain improves the appearance in the frequency domain as this is essentially smoothing the spectrum with the response of the window.
Remember if you don’t use a window you have a rectangular window by default which recall has a sinc response in the frequency domain, with quite big sidelobes.
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