Maintaining the Hermitian Property

The input signal in the Fourier transform in general is complex. In this application the volume data in spatial domain is stored only in the real component and the imaginary component is set to zero. This, so called real function, is a Fourier transform pair to a Hermitian function. Therefore a Fourier transform of this spatial domain representation of the volume leads to a frequency domain representation which is Hermitian. The following equations clarify the meaning of the Hermitian property.

$$\hat{X}_N[\kappa] = \hat{X}_N[-\kappa] \ast$$ (5.3)

$$Re\{\hat{X}_N[\kappa]\} = Re\{\hat{X}_N[-\kappa]\}$$ (5.4)

$$Im\{\hat{X}_N[\kappa]\} = -Im\{\hat{X}_N[-\kappa]\}$$ (5.5)

Corresponding sample values, with index $\kappa$ and $-\kappa$, are conjugate complex to each other, the real components are equal (Equation 5.4), and the imaginary components have inverse sign (Equation 5.5).

Figure: (a) A real function in spatial domain is transformed to a (b) Hermitian function in the frequency domain. If the real and the imaginary component of the Hermitian function are explored independently, then the real component forms an even function and the imaginary component an odd function.

(a)	(b)

Another way to look at the Hermitian property is to explore the real and imaginary component independently (see Figure 5.2). The complex function in Figure 5.2(b) consists of an even function in the real component, and an odd function in the imaginary component.

Figure: Hermitian relation of functions with an even (a) and an odd (b) number of samples. The arrows indicate which samples have to be conjugate complex to each other to maintain the Hermitian property.

(a)	(b)

How the Hermitian relation applies to the indexing scheme in the discrete case is presented in Figure 5.3, there are two different patterns for even and odd number of samples. The reason for going into details about Hermitian functions is that they are dual to the real function by the Fourier transform. That means all operations applied in the frequency domain have to preserve the Hermitian property in order to obtain a real function after the inverse Fourier transform. The indices which need special attention are listed in Table 5.1. The first issue treats sample points that are conjugate complex to itself, and therefore the imaginary component has to be zero. That is taken care of by the Fourier transform itself, the result after the transformation to the frequency domain contains zeros at the correct positions. Some available packets like FFTW [9] have special real Fourier transforms that exploit the symmetry in the data to save roughly a factor of two in both time and storage. The interface for this special transform does not allocate memory for these zero indices, to save storage space. The second issue has an effect when calculating the derivative, Section 4.4.7. If the length $N$ of the signal $\hat{X}_N[\mu]$ is even with $N=2\beta$ then $\hat{X}_N[\beta]$ has to be conjugate complex to itself to maintain Hermitianity (see Figure 5.3(a)). The multiplication with $i\mu 2\pi \frac{1}{N}$ essentially switches the real and imaginary component. As the result still has to be Hermitian, $i\mu2\pi \frac{1}{N}\hat{X}_N[\beta]$ again has to be conjugate complex to itself. This forces $Re\{\hat{X}_N[\beta]\}$ and $Im\{\hat{X}_N[\beta]\}$ to be zero. Another way of getting around this problem is to add one zero valued sample at the end of the signal in spatial domain, in order to create an odd number of samples (spatial domain zero padding).

Table 5.1: Problematic indices for the Hermitian property

	index	problem	counter measure
function	$\kappa = 0$ $\kappa = n$ if N is even	$\hat{X}_N[\kappa] = \hat{X}_N[\kappa] \ast$	$Im\{\hat{X}_N[\kappa]\} = 0$
derivative	$\kappa = n$ if N is even	$i\hat{X}_N[\kappa] = (i\hat{X}_N[\kappa]) \ast$	$Re\{\hat{X}_N[\kappa]\} = 0$ $Im\{\hat{X}_N[\kappa]\} = 0$