Frequency array feed real values FFT

Question

EDIT

The first version of my question has not worked as I expected, so that I will try to be a little bit more specific.

The final goal I am trying to achieve is the generation of a ten minutes time series: to achieve this I have to perform an FFT operation, and it's the point I have been stumbling upon.

Generally the aimed time series will be assigned as the sum of two terms: a steady component $U(t)$ and a fluctuating component $u^{'}(t)$. That is

$$u(t) = U(t) + u^{'}(t);$$

So generally, my code follows this procedure:

1) Given data

$time = 600 [s];$

$Nfft = 4096;$

$L = 340.2 [m];$

$U = 10 [m/s]$

$df = 1/600 = 0.00167 Hz;$

$f_{n} = Nfft/(2*time) = 3.4133 Hz;$

This means that my frequency array should be laid out as follows:

$$ f = (-f_{n}+df):df:f_{n} $$

But, instead of using the whole $f$ array, I am only making use of the positive half:

$$ f_{+} = df:f_{n} = 0.00167:3.4133 Hz; $$

2) Spectrum Definition

I define a certain spectrum shape, applying the following relationship

$$ S_{u} = \frac{6L/U}{(1 + 6f_{+}L/U)^{5/3}}; $$

3) Random phase generation

I, then, have to generate a set of complex samples with a determined distribution: in my case, the random phase will approach a standard Gaussian distribution $(\mu = 0, \sigma = 1)$.

In MATLAB I call

nn = complex(normrnd(0,1,Nfft/2),normrnd(0,1,Nfft/2));

4) Apply random phase

To apply the random phase, I just do this

$$ H_{u} = S_{u}*nn; $$

At this point start my pains!

So far, I only generated $Nfft/2 = 2048$ complex samples accounting for the $f_{+}$ content. Therefore, the content accounting for the negative half of $f$ is still missing. To overcome this issue, I was thinking to merge the $real$ and $imaginary$ part of $H_{u}$, in order to get a signal $H_{uu}$ with $Nfft = 4096$ samples and with all real values.

But, by using this merging process, the $0-th$ frequency order would not be represented, since the $complex$ part of $H_{u}$ is defined for $f_{+}$.

Thus, how to account for the $0-th$ order by keeping a procedure as the one I have been proposing so far?

Answer 1

There is still lots of issues with what you're trying to achieve, but it's a little clearer for me now. Thanks for the edit.

1. DC term: You say $$ u(t) = U(t) + u^{'}(t); $$ with $U = 10 [m/s]$. Surely the constant $U$ is your "DC" (zeroth order) term?

2. Sampling Rate: You say that $time = 600$ and that $Nfft = 4096$. Does that make your sampling rate $f_s = 4096 / 600 = 6.8267$ Hz?

3. df Choice: I am not sure why you choose $df = 600$ ?

4. Random Phase Generation: Your phase generation seems odd to me. This nn = complex(normrnd(0,1,Nfft/2),normrnd(0,1,Nfft/2)); will not generate just random phase. It will also generate a random amplitude, which is not what you want if you're just after phase. To get a random phase, you're better off doing: nn = exp(1j*2*pi*rand(1,Nfft/2)). That generates Nfft/2 uniform random variables (between 0 and 1), multiplies them by $2\pi$ and then forms $e^{j2\pi\times \mathrm{rand(1,Nfft/2)}}$.

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How to account for the DC term?

If your (positive-frequency) spectrum is $H_n(f)$ for $f= k df$ for $k=1,\ldots,Nfft/2$, then just form:

$$ \begin{array} \ H(f) &=& U, \mathrm{for\ } f=0\\ &=& H_n(f), \mathrm{for\ } f>0\\ &=& H^*_n(-f), \mathrm{for\ } f<0 \end{array} $$

Answer 2

To get a real signal valued time domain signal: create the negative frequencies as the complex conjugate of the positive frequencies

To get a complex time domain signal: create negative frequencies by copying the amplitudes of the positive frequencies and applying a random phase.

If you construct the signal in frames you will been apply windowing and overlap to avoid discontinuities at the frame boundaries.