Why do we get ripples in the frequency spectrum of a chirp, and how do I remove it?

Question

The ripples in the amplitude spectrum of a linear spectrum is undesirable. I desire to obtain a flat spectrum response. Hence, how can I remove these ripples? Also, which factors (like time bandwidth product, chirp rate) are affecting the ripples, and how?

I designed a chirp with a frequency from 10 to 90 Hz in MATLAB. However, I obtained Fresnel ripples (which normally occurs) in the passband, along with overshoots at the band edges. I need to study the application of chirps on a material and use it for defect analysis. For this, I need to optimize the whole energy applied to the chirp. Hence, I want to get rid of these ripples. So, what parameters like chirp rate and time of sweep will affect the system?

Answer 1

Another thing the OP can do is use the "complex" chirp. What you really have to do is (at two different times and both times sweeping all frequencies from -Nyquist through DC to +Nyquist) pass the real part, $x_r(t)=\cos(\pi \beta t^2)$ and then pass the imaginary part $x_i(t)=\sin(\pi \beta t^2)$ of the complex chirp through the material under test and synchronously measure the two real results ($y_r(t)$ and $y_i(t)$), then assemble them as the real and imaginary parts of the response.

Let $$ \begin{align} x(t) & = e^{j \pi \beta t^2} \\ & = \cos(\pi \beta t^2) + j \sin(\pi \beta t^2) \\ & = x_r(t) \qquad + j \ x_i(t) \\ \end{align} $$

Then $$ X(f) \ \triangleq \ \mathscr{F}\left\{ x(t) \right\} \ = \ \sqrt{\frac{j}{\beta}} e^{-j\frac{\pi}{\beta} f^2} $$

You will see that $ |X(f)| = \sqrt{\frac{1}{\beta}} $ for all $f$.

The respective outputs that are synchronously measured with the two driving signals are $y_r(t)$ and $y_i(t)$:

$$ \begin{align} y_r(t) & \triangleq h(t) \circledast x_r(t) \\ & = h(t) \circledast \cos(\pi \beta t^2) \\ \\ y_i(t) & \triangleq h(t) \circledast x_i(t) \\ & = h(t) \circledast \sin(\pi \beta t^2) \\ \end{align} $$

Then this aggregate complex output, $y(t)$, can be assembled (in the mind of the computer) from the responses of the cosine sweep (the real part) and the sine sweep (the imaginary part):

$$ \begin{align} y(t) & = y_r(t) + j \ y_i(t) \\ & = h(t) \circledast x_r(t) + j \ h(t) \circledast x_i(t) \\ & = h(t) \circledast \left(x_r(t) + j \ x_i(t)\right) \\ & = h(t) \circledast x(t) \\ & = \int_{-\infty}^{+\infty} h(u) \, x(t-u) \ du \\ & = \int_{-\infty}^{+\infty} h(u) \, e^{j \pi \beta (t-u)^2} \ du \\ & = \int_{-\infty}^{+\infty} h(u) \, e^{j \pi \beta (t^2-2tu+u^2)} \ du \\ & = e^{j \pi \beta t^2} \int_{-\infty}^{+\infty} \left(h(u) \, e^{j \pi \beta u^2}\right) \, e^{-j 2 \pi \beta t u} \ du \\ & = e^{j \pi \beta t^2} \int_{-\infty}^{+\infty} \tilde{h}(u) \, e^{-j 2 \pi (\beta t) u} \ du \\ & = e^{j \pi \beta t^2} \tilde{H}(\beta t) \\ \end{align} $$

where $$ \tilde{H}(f) \triangleq \int_{-\infty}^{+\infty} \tilde{h}(t) e^{-j 2 \pi f t} \ dt = \mathscr{F}\left\{\tilde{h}(t) \right\} $$

and $$ \tilde{h}(t) \triangleq h(t) e^{j \pi \beta t^2} $$

So this is what you get: first consider making this adjustment (dechirping) in the measured output: $$ y(t) e^{-j \pi \beta t^2} = \tilde{H}(\beta t) $$

Now if the sweep rate $\beta$ is low enough, then $e^{j \pi \beta t^2} \approx 1$ for as long as $h(t)$ is not close to 0. Then $\tilde{h}(t) \approx h(t)$ and $\tilde{H}(f) \approx H(f)$ and $\tilde{H}(\beta t) \approx H(\beta t)$ and what you have measured in the output is another chirp, but with relative magnitude and phase changed by $\tilde{H}(\beta t) \approx H(\beta t)$.

Even if the sweep rate is not so low, you can compute either the true frequency response $H(f)$ by first computing

$$ \begin{align} \tilde{h}(t) & = \mathscr{F}^{-1}\left\{ \tilde{H}(f) \right\} \\ & = \int_{-\infty}^{+\infty} \tilde{H}(f) e^{j 2 \pi f t} df \\ & = h(t) e^{j \pi \beta t^2} \\ \end{align} $$

Or from that result you can get the true impulse response $h(t)$ and then $H(f)$ if that's what you want. It should be the same $H(f)$ you would get if you divided $Y(f) = \mathscr{F}\left\{ y(t) \right\} $ by $X(f) = \mathscr{F}\left\{ x(t) \right\} = \sqrt{\frac{j}{\beta}} e^{-j\frac{\pi}{\beta} f^2}$. And there is no division by zero, because there are no ripples in $X(f)$ because the magnitude is a constant $\sqrt{\frac{1}{\beta}}$.

That is how to remove ripples (and any danger of division by zero) in the spectrum of a linearly-swept chirp.

Answer 2

Depends on your application constraints. Easiest way would be the following

1. Create chirp in the time domain
2. Fourier Transform
3. Set the spectrum amplitude to what you want it to be
4. Inverse Fourier Transform

This produces a chirp-like time domain signal that has the exact frequency spectrum that you want

Answer 3

I might be wrong, but my guess is that these ripples appear as a result of your chirp function being finite in time, generating discontinuities at both ends of your signal. The classic way to reduce ripples caused by discontinuities is multiplying your signal by a window, making the transition smoother. In Matlab you could try something like:

smoothChirp = chirp.*hamming(l);

Where l is the length of your signal. Of course the spectrum of your chirp won't have a "flat" characteristic anymore, so in the end we have a trade-off between perfectly linear frequency variation and the appearance of ripples in the signal spectrum.