The Fourier Doppelgangers

It is well known that Fourier Transform is unique under certain conditions that are satisfied by almost all practical signals. Then, how can we resolve the following contradiction?

Consider a sinc pulse and Linear Frequency Modulated (LFM) pulse (a chirp) in time domain.

• The sinc pulse is defined as
  \[
  \text{sinc}(t) = \frac{sin(\pi t)}{\pi t}
  \]

  Now the spectrum of a sinc pulse in time in an ideal case is a rectangular signal in frequency domain, which is the most fundamental relation in signal processing. Both the sinc pulse and its spectrum are plotted in the left half of the figure below.



• A linear frequency modulated pulse is given by
  \[
  p(t) = \exp\left[j(\pi \mu t^2 + 2\pi f_0t + \phi_0)\right],
  \]

  where $\mu$ is the chirp rate, $f_0$ is the starting frequency and $\phi_0$ is the initial phase. Instead of a linear increase in phase (as in a sinusoid), an LFM pulse has a phase that increases quadratically with time.
  Both the LFM pulse and its spectrum are plotted in the right half of the figure above. The spectrum of an LFM pulse in an ideal case is a rectangular signal in frequency domain (imagine the signal starting from $F_0$ and ending at $F_1$, with its time support extended infinitely in both directions). Why? Because a linear sweep across all frequencies within the bandwidth $B$ evenly spreads the energy across the whole spectrum.

The question is: How can two different signals have a similar spectrum? Can there be Doppelgängers in frequency domain?

The key to answering this question is the Magnitude label on y-axis of bottom subplots. Both of these pulses have identical magnitude spectra (only in the limit time-bandwidth product goes to infinity, but plotting them from finite duration signals makes the question better than plotting two perfect rectangles at the bottom), but not the same phase spectra.

• A sinc pulse has a linear phase, as its coefficients are symmetric around the center.
• The LFM pulse has a quadratic phase due to the term $\pi \mu t^2$.

In conclusion, the question only focuses on the magnitude, and this is the whole point. This is the first lesson one should learn when entering the real world. A question can have insufficient information or can even be wrong. In fact, anything can be wrong. And that posing the right questions is much more important than finding the right answers. In this case, while magnitude spectrum is the center of attention in most cases, phase spectrum is also important in many applications.