Problem set — the Fourier transform and convolution

Problem set: the Fourier transform and convolution¶

Exam-style problems. Work them with a pen before touching a computer; the last three are meant to be finished numerically. Solutions and marking rubrics live with the instructor material and are deliberately not on this site.

Throughout, the course convention holds: the forward transform is $F(\omega) = \int f(t)\, e^{-\ii\omega t}\,\mathrm{d}t$, and the inverse carries the opposite sign together with the $1/2\pi$ — see the conventions page. RMS widths are second moments of $|f|^2$ and $|F|^2$ about their centroids.

Problem 1 — the Gaussian pair, honestly¶

(a) Transform $f(t) = e^{-t^2/2\sigma^2}$ by completing the square in the exponent, and obtain $F(\omega) = \sqrt{2\pi}\,\sigma\,e^{-\sigma^2\omega^2/2}$. State clearly where you used the standard Gaussian integral and why the shift into the complex plane is legitimate.

(b) Compute $\Delta t_{\text{rms}}$ and $\Delta\omega_{\text{rms}}$ for this pair. Show that each is smaller than the width of the curve it describes by a factor of $\sqrt{2}$, and that their product is exactly $\tfrac12$.

(c) Explain, without proving it, why equality singles out the Gaussian: what is special about a function whose transform has the same shape?

Problem 2 — the theorems that move things¶

(a) Prove the scaling theorem: if $g(t) = f(at)$ for $a > 0$, then $G(\omega) = F(\omega/a)/a$. Say in one sentence what happens to the area under $|F|$ and why that had to be so.

(b) Prove the shift theorem: delaying a signal by $t_0$ multiplies its transform by $e^{-\ii\omega t_0}$. Deduce that $|F(\omega)|$ is completely blind to when a signal happened, and comment on what that implies for reconstructing a signal from its magnitude alone.

(c) Prove the modulation theorem: multiplying by $\cos\omega_c t$ splits the spectrum into two copies at $\pm\omega_c$, each at half height. This is amplitude modulation, and it is how the radio stations of the opening puzzle share the air.

(d) Using only the zoo and these three theorems — no integrals — sketch the spectra of: a rectangular pulse of width $a$ delayed by $3a$; a Gaussian multiplied by $\cos\omega_c t$; and two rectangular pulses of width $a$ separated by $d$.

Problem 3 — linewidths and lifetimes¶

(a) For the one-sided exponential $e^{-t/\tau}\,\Theta(t)$, show that $|F(\omega)|^2$ is a Lorentzian and that its full width at half maximum is $2/\tau$.

(b) An excited atomic state decays with a lifetime of $2.0\ \mathrm{ns}$. What is the minimum linewidth of the light it emits, in MHz? Compare this with the Doppler width of a room temperature gas and say which dominates.

(c) Use Parseval to evaluate $\int_{-\infty}^{\infty} \mathrm{d}\omega/(1/\tau^2 + \omega^2)$ without contour integration.

(d) A measured line is the true lineshape convolved with the instrument’s response. If both are Lorentzian, what is the shape and width of the result? If both are Gaussian? (One of these answers generalises easily and the other does not — say which.)

Problem 4 — hunting the minimum product, numerically¶

Using wavelab.fourier (in the browser laboratory or locally):

(a) Build the supergaussian family $f(t) = \exp(-|t/\sigma|^{p})$ for $p$ from 1 to 8, and compute $\Delta t_{\text{rms}}\Delta\omega_{\text{rms}}$ for each with rms_widths. Plot the product against $p$.

(b) Confirm the minimum sits at $p = 2$ with value $\tfrac12$, and that every other member of the family is above it. Confirm also that the product does not change when $\sigma$ does.

(c) As $p$ grows the pulse approaches a rectangle. Explain what happens to the measured product, and why a hard-edged rectangle is a badly behaved member of this family. (Consider how the sinc tails contribute to the second moment of $|F|^2$.)

Problem 5 — leakage on trial¶

Using wavelab.fourier:

(a) Take a pure cosine whose frequency lands exactly on a grid frequency of your record, and compute its spectrum. Then shift the frequency by half a grid spacing and recompute. Report the peak height in both cases and describe where the missing energy went.

(b) Repeat with a Hann window applied to the record. Report the sidelobe level and the main-lobe width in both cases, and state the trade in one sentence.

(c) Show that zero-padding the record by a factor of eight makes the plotted spectrum smoother but does not separate two tones that were unresolved before. Then find, by experiment, the record length that does separate two tones 1.5 grid spacings apart, and compare it with $2\pi/\Delta\omega$.

Problem 6 — challenge: putting the samples back together¶

(a) Implement sinc interpolation: reconstruct a continuous signal from samples taken at spacing $\Delta t$ by summing $\operatorname{sinc}$ functions centred on each sample. Verify that for a signal band-limited below Nyquist the reconstruction is exact to machine precision at points between the samples, not merely at them.

(b) Break it. Feed the same routine a chirp that sweeps up through the Nyquist frequency, and plot reconstruction against truth. Explain the failure in terms of the comb picture, and say precisely at what moment the reconstruction stops being recoverable.

(c) Take any signal, keep $|F(\omega)|$, replace every phase by a random one, and invert. Compare with the original. Then do the reverse — keep the phases and flatten every magnitude to one — and compare again. Which reconstruction is more recognisable, and what does the pair of experiments say about where a signal’s information actually sits?