Why does spectrum magnitude decay away from DC for positive signals?

Question

I would like to understand why the second frequency component usually has the greater magnitude within the range [1, (N/2)], i.e why (I remove the DC component and the complex conjugates freq after N/2) when we're computing the FFT of a positive real-valued signal. Can I have a mathematical explanation of this phenomenon?

I think it's related to the impact of m within : . But I'm unable to identify how it impacts the magnitude of the complex number X[m] compared to X[m+1].

Answer 1

It's not.

Consider:

$$x_1[n] = \cos\left(\frac{2\pi n}{8}\right)$$

which is real-valued and now let's take its absolute value (to make it strictly positive) so now we have:

$$x[n] = \left|\cos\left(\frac{2\pi n}{8}\right)\right|$$

which looks like this for $n = 0, \cdots , 15$

And here is the plot of the real and imaginary parts of its 8-Pt DFT along with the Magnitude:

As you can see $X[1] < X[2]$ (indexed with 2 and 3 in the plots due to Matlab's indexing convention).

Answer 2

No such property exists in the general case. In Ahsan Yousaf's answer a counter-example was shown. However, that example is very special in the sense that the sequence is periodic (inside the length-$N$ window), and, consequently, the DFT must have zeros at regular intervals.

One might suppose then that in the case when there is no periodicity in the time domain, the property assumed by the OP may still hold. However, that's also not the case. Take for example the non-periodic sequence

x = [.9,.95,.75,.05,.55,.8,.55,.2]

The figure below shows the magnitude of the DFT $X[k]$ of the sequence $x[n]$:

Clearly, $|X[k]|$ is not monotonic.

There is, however, a related property that always holds for non-negative sequences:

$$|X[k]|\le X[0],\quad k=1,\ldots,N-1, \quad x[n]\ge 0$$

which can be shown as follows:

$$\big|X[k]\big|=\left|\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\right|\le \sum_{n=0}^{N-1}\big|x[n]\big|=\sum_{n=0}^{N-1}x[n]=X[0]$$

Answer 3

Positivity doesn't affect spectrum whatsoever, except for the DC bin - try adding $1$ to a 1-normed audio file. However, such behavior is observed with absolute value of bandpass signals (complex) - but same as with the example you provide, it's not strict for all k > k + 1, rather more approximate. Strict k > k + 1 is observed for smoothly-decaying signals (including real-valued).

This has important applications in time-frequency analysis, particularly helpful for machine learning or audio synthesis, due to its induced sparsity, and it's heavily exploited by Wavelet Scattering.

Example

Why does it happen? In time domain, we observe that the envelope is less likely to vary as quickly as the real or imaginary parts, since both parts in $\Re e^2 + \Im m^2$ would have to vary fast together. In frequency, we have $|x|^2 \Leftrightarrow |X|^2$, which is a convolution of $X$ with its own conjugate, which simply doubles the spectrum while keeping it bandlimited. If we treat $\sqrt{|X|^2}$ as a rough approximation, the behavior is roughly reproduced. Also, bandwidth of $|\sin(x)|$ is roughly doubled.

Why [1:]? Because the DC bin of a positive signal is by far the most dominant, in vast majority of cases. Recall, DC = sum, so every single point only increases it. It's also guaranteed $X[0] \geq |X[k]|$ for all $k$ (see Matt's answer).

Strict examples

So, generally but not always, Monotonic variation${}^1$ in time $\Rightarrow$ Monotonic decay in freq (note, not $\Leftrightarrow$). This isn't a surprise, since for many decay behaviors, this result holds exactly:

$$ \begin{align} 1/\sqrt{|t|} &\Leftrightarrow 1/\sqrt{|\omega|} \\ 1/|t| &\Leftrightarrow -\log{|\omega|} + C \\ 1/t^2 &\Leftrightarrow -\omega\ \text{sgn}(\omega) \\ e^{-t^2} &\Leftrightarrow e^{-\omega^2} \end{align} $$

The converse, "Monotonic decay in freq $\Rightarrow$ Monotonic variation in time", doesn't hold; scrambling the phase of such a spectrum, by multiplying with unit-magnitude noise, will convolve with a scrambled kernel in time, destroying monotonicity without changing $|X|$. Note, the cosine, $-t$, and $\texttt{floor}\{-t\}$ examples are only valid in discrete - they're just to show different decay behaviors.

1: "variation" meaning decay or growth, and it doesn't have to occur over the full duration (e.g. any Gaussian works). All above plots could flip the signal and obtain a near-identical |FFT| - or identical flipping via x[1:] = x[1:][::-1].