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Can the Fourier Transform Magnitude Be Negative?

February 16, 2022

Introduction

I came across a question on DSP Stack Exchange the other day: can the Fourier Transform magnitude be negative? This is a great question! The answer is no, and let’s take a look at why.

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Discrete Time Fourier Transform

The discrete-time Fourier transform (DTFT) is

(1) $\begin{equation*}X\left(e^{j\omega}\right) = \sum_{n=-\infty}^{\infty} x[n] e^{-j2\pi \frac{f}{f_s} n}.\end{equation*}$

The DTFT is a cross correlation of x[n] with a complex exponential $e^{j2\pi \frac{f}{f_s}n}$ . The summation over all n produces a single complex number. The DTFT is written as a real and imaginary component,

(2) $\begin{equation*}X\left(e^{j\omega}\right) = X_{R}\left(e^{j\omega}\right) + j X_{I}\left(e^{j\omega}\right),\end{equation*}$

where

(3) $\begin{equation*}X_{R}\left(e^{j\omega}\right) = \text{RE} \left\{ X\left(e^{j\omega}\right) \right\}\end{equation*}$

and

(4) $\begin{equation*}X_{I}\left(e^{j\omega}\right) = \text{IM} \left\{ X\left(e^{j\omega}\right) \right\}.\end{equation*}$

Magnitude of Complex Number

Rather than use the complex number in (2) a simpler representation is given by

(5) $\begin{equation*}c = a + jb\end{equation*}$

where the complex number c has a real part a and an imaginary part jb. The magnitude of a complex number c can be written as

(6) $\begin{equation*}|c| = \sqrt{c \cdot c^*}.\end{equation*}$

The magnitude can be written in terms of a and b by substituting (5) into (6),

(7) $\begin{equation*}\begin{split}|c| & = \sqrt{(a + jb) \cdot (a + jb)^*} \\& = \sqrt{(a + jb) \cdot (a - jb)} \\& = \sqrt{a^2 - jab + jab + b^2} \\& = \sqrt{a^2 + b^2}.\end{split}\end{equation*}$

The square of a real number $a^2$ and $b^2$ cannot be negative and the square root of a positive number is always positive, therefore the magnitude $|c|$ cannot be negative.

Plotting Magnitude of Complex Number

Conceptually the magnitude of a complex number c is the length of the vector beginning at the origin 0 + 0j and ending at c. Consider a complex number

(8) $\begin{equation*}c = -3 - 7j.\end{equation*}$

Figure 1 plots the complex vector c in 2D complex space.

The magnitude $|c|$ is the distance from 0+0j to -3-7j which is

(9) $\begin{equation*}\begin{split}|c| & = \sqrt{(-3)^2 + (-7)^2} \\& = \sqrt{ 9 + 49 } \\& = 7.61\end{split}.\end{equation*}$

Fourier Transform Magnitude

Using the representation in (2) for the complex number the magnitude can be written as

(10) $\begin{equation*}|X\left(e^{j\omega}\right)| = X_{R}\left(e^{j\omega}\right)^2 + X_{I}\left(e^{j\omega}\right)^2\end{equation*}$

which is always a positive number.

Conclusion

The magnitude of a complex number always has to be positive by definition, it can never be negative. The magnitude is conceptually the length of a complex vector which is written mathematically as $\sqrt{ a^2 + b^2}$ .

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Got a question? Drop it in the comments below.

One Response

Chad M Spooner says:

March 3, 2022 at 7:49 pm

I definitely agree that the magnitude of the Fourier transform, or any complex number, is non-negative. But students or newcomers to DSP might be a bit confused about this statement: “the square root of a positive number is always positive.” Because square roots of positive numbers can be negative: -2 is a valid square root of 4. I think the difference is in the definition of “magnitude” (or “modulus” or “absolute value”) which is defined as the positive square root.

When we use the square-root symbol, and there is no ‘-‘ or ‘+’ in front of it, ‘+’ is assumed.

Interested readers might look into metric spaces and inner products:

https://en.wikipedia.org/wiki/Magnitude_(mathematics)

https://en.wikipedia.org/wiki/Euclidean_space#Euclidean_norm

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