dsp – Page 7 – Daniel Estévez

Improving signal processing in my OTH radar receiver

This is a follow up post to my experiments studying OTH radar. I have found that the signal processing I did there to obtain the cross-correlation was far from optimal. This produced the strange side-bands below the main reflection. The keen reader might have noticed that I was doing the cross-correlation with a template pulse that lasted the whole pulse repetition cycle. However, the pulses from the radar are shorter. Therefore, it is a better idea to use a shorter template pulse. Ideally, the template pulse should have the same length as the transmitted pulse. However, I don’t know this length precisely, because multipath propagation makes the received pulses a bit longer. However, I think that 6.5ms is a good estimate.

I have also changed the window for the pulse. I’m now using a Dolph-Chebyshev window. I use scipy to compute this window, because it is not included in GNU Radio. This window has the property that the side-bands have constant attenuation. Indeed, it minimizes the \(L^\infty\) norm of the side-bands. There is a parameter that tunes the side-bands attenuation. For higher attenuations, you have a wider main lobe, while if you want a narrower main love you get less side-band attenuation. These properties make this window useful in radar applications.

Below I’m doing the cross-correlation in GNU Radio with a shorter template pulse shaped with a Dolph-Chebyshev window set for 55dB attenuation.

The good thing about the settable attenuation of the Dolph-Chebyshev window is that it can be used to trade-off performance between different features. First, we use an attenuation of 100dB. The side-bands are below the noise floor in this case, so we have no “false responses” in our cross-correlation. The drawback is that the main lobe is wide so the resolution of the features of the ionosphere in the image below is not very good.

Dolph-Chebyshev window with 100dB attenuation

Next we try with 55dB attenuation. This narrows the main lobe, improving the resolution and crispness of the features of the ionosphere in the image below. However, side-bands start being visible above the noise floor, producing “false responses”. However, comparing with the image above, we now know where the false responses are.

Dolph-Chebyshev window with 55dB attenuation

I have updated the gist with the GNU Radio flowgraph and python script used to produce the images.

Scramblers and their implementation in GNUradio

A scrambler is a function that is applied to a sequence of data before transmitting with the goal of making this data more “random-like”. For instance, scrambling avoids long runs of the bits 0 or 1 only, which may make the receiver loose synchronization or cause spectral concentration of the signal. The receiver applies the inverse function, which is called descrambler, to recover the original data. The documentation for the scrambler blocks in GNUradio is not very good and one may need to take a look at the implementation of these blocks to get their parameters right. Here, I’ll try to explain a bit of the mathematics behind scramblers and the peculiarities of their implementation in GNUradio.

Another look at recursive quadrature oscillators

In a recent post, we looked at which \(2\times 2\) Toeplitz real matrices \(T\) gave useful quadrature oscillators by the recurrence \(x_{n+1}=T x_n\). There, we computed their eigenvalues and solved the recurrence in terms of them. Of course, there are many other ways to approach this problem. Here we look at another approach that gives a good geometric picture of what happens, can be applied to general \(2\times 2\) matrices, and may be used as a starting point for the \(n\times n\) case.

A look at a new digital quadrature oscillator

Two sinusoidal signals are said to be in quadrature if they have a constant phase difference of 90º. Quadrature signals are widely used in signal processing. A digital quadrature oscillator is just an algorithm that computes the sequence \(x_n = (\cos(\omega n), \sin(\omega n))\), \(n\geq 0\), or a similar sequence of sinusoids in quadrature. Here \(\omega\) is the oscillator frequency in radians per sample. Direct computation of this sequence is very time consuming, because the trigonometric functions have to be evaluated for each sample. Therefore, it is a good idea to use a linear recurrence scheme to compute \(x_n\). Using basic trigonometric identities, we see that\[x_{n+1} = A x_n,\quad x_0=\begin{pmatrix}1\\0\end{pmatrix},\]with\[A = \begin{pmatrix}\alpha_1 & -\alpha_2\\\alpha_2 & \alpha_1\end{pmatrix},\quad \alpha_1 = \cos(\omega),\ \alpha_2=\sin(\omega).\]

However, to actually perform these computations in a digital processor, one has to quantize \(\alpha_1,\alpha_2\), meaning that one has to replace \(\alpha_1,\alpha_2\) by approximations. It is easy to see that if one replaces \(\alpha_1,\alpha_2\) by some perturbation, then the eigenvalues of \(A\) are no longer in the unit circle, so the oscillation can grow or decay exponentially and one would need to apply an AGC scheme to keep this method stable.

Here we will look at a new quadrature oscillator by Martin Vicanek that has appeared recently and solves this problem.