# Symmetries of Second-Order Probabilistic Parameters in CSP

As you progress through the various stages of learning CSP (intimidation, frustration, elucidation, puzzlement, and finally smooth operation), the symmetries of the various functions come up over and over again. Exploiting symmetries can result in lower computational costs, quicker debugging, and easier mathematical development.

What exactly do we mean by ‘symmetries of parameters?’ I’m talking primarily about the evenness or oddness of the time-domain functions in the delay $\tau$ and cycle frequency $\alpha$ variables and of the frequency-domain functions in the spectral frequency $f$ and cycle frequency $\alpha$ variables. Or a generalized version of evenness/oddness, such as $f(-x) = g(x)$, where $f(x)$ and $g(x)$ are closely related functions. We have to consider the non-conjugate and conjugate functions separately, and we’ll also consider both the auto and cross versions of the parameters. We’ll look at higher-order cyclic moments and cumulants in a future post.

You can use this post as a resource for mathematical development because I present the symmetry equations. But also each symmetry result is illustrated using estimated parameters via the frequency smoothing method (FSM) of spectral correlation function estimation. The time-domain parameters are obtained from the inverse transforms of the FSM parameters. So you can also use this post as an extension of the second-order verification guide to ensure that your estimator works for a wide variety of input parameters.

### The Non-Conjugate Cyclic Autocorrelation Function

Let’s start our symmetry odyssey with arguably the simplest CSP function: the non-conjugate cyclic autocorrelation function. Recall that this function is equal to the conventional autocorrelation function when the cycle frequency $\alpha$ is zero.  The non-conjugate cyclic autocorrelation is defined by

$\displaystyle R_x^\alpha(\tau) = \left\langle x(t+\tau/2)x^*(t-\tau/2) e^{-i2\pi\alpha t} \right\rangle_t \hfill (1)$

where the notation $\left\langle \cdot \right\rangle_t$ denotes the infinite time average:

$\displaystyle \left\langle x(t) \right\rangle_t = \lim_{T\rightarrow\infty} \frac{1}{T} \int_{-T/2}^{T/2} x(t) \, dt. \hfill (2)$

The symmetry in the lag parameter $\tau$ is determined by examining $\displaystyle R_x^\alpha(\tau)$ for negative $\tau$,

$\displaystyle R_x^\alpha(-\tau) = \left\langle x(t-\tau/2)x^*(t+\tau/2) e^{-i2\pi\alpha t} \right\rangle_t \hfill (3)$

A little algebra yields

$\displaystyle R_x^\alpha(-\tau) = \left[ \left\langle x(t+\tau/2)x^*(t-\tau/2) e^{-i2\pi(-\alpha) t} \right\rangle_t \right]^* = R_x^{-\alpha}(\tau)^* \hfill (4a)$

Note that (4) also reveals the symmetry in the cycle frequency parameter, since it can be rearranged as

$\displaystyle R_x^\alpha(\tau) = R_x^{-\alpha}(-\tau)^*. \hfill(4b)$

Let’s illustrate our sequence of symmetry results using our old friend, the rectangular-pulse BPSK signal. It has ten samples per bit, or $f_{bit} = 0.1$ Hz, and a carrier frequency offset of $0.05$ Hz. The carrier frequency offset renders the signal complex valued, and we also add complex-valued white Gaussian noise. The power of the signal is unity and the power of the noise is $0.1$. The true parameters for this signal are known: frequency domain and time domain.

First, let’s take a look at the autocorrelation, which yields the symmetry relation (4) with $\alpha = 0$,

$\displaystyle R_x^0(-\tau) = R_x^0(\tau)^* \hfill (5)$

This implies that the real part and the magnitude are even functions of $\tau$ and the imaginary part is an odd function of $\tau$. Figure 1 shows the estimated autocorrelation, confirming these even/odd predictions:

Moving on to the cycle frequency of $\alpha = f_{bit} = 0.1$, we obtain the estimates shown in Figure 2. The result in (4) implies that the real part of the CAF for $\alpha = f_{bit}$ at $\tau$ should be equal to the real part of the CAF for $\alpha = -f_{bit}$ at $-\tau$ (verify with $\tau = 4$). It also implies that the imaginary part for $\alpha$ and $\tau$ should be the negative of that for $-\alpha$ and $-\tau$ (verify with $\tau = -6$). So the equation and the estimates are consistent.

### The Conjugate Cyclic Autocorrelation Function

A similar analysis applies to the conjugate cyclic autocorrelation function. Let’s go through it. The function is defined by

$\displaystyle R_{x^*}^\alpha(\tau) = \left\langle x(t+\tau/2)x(t-\tau/2) e^{-i2\pi \alpha t} \right\rangle_t \hfill (6)$

Symmetry in $\tau$ follows easily

$\displaystyle R_{x^*}^\alpha(-\tau) = R_{x^*}^\alpha(\tau). \hfill (7)$

because we don’t have any conjugations to deal with. This simple result implies that the real and imaginary parts (and the magnitude) are even functions of $\tau$. This symmetry is validated using estimates in Figure 3.

The symmetry relation (7) tells us nothing about the symmetry in cycle frequency $\alpha$. So let’s look at $R_x^{-\alpha}(\tau)$,

$\displaystyle R_{x^*}^{-\alpha}(\tau) = \left\langle x(t+\tau/2) x(t-\tau/2)e^{-i2\pi(-\alpha)t} \right\rangle_t \hfill (8)$

$\displaystyle = \left\langle \left[ x^*(t+\tau/2)x^*(t-\tau/2) e^{-i2\pi\alpha t} \right]^* \right\rangle_t \hfill (9)$

$\displaystyle = \left\langle x^*(t+\tau/2)x^*(t-\tau/2) e^{-i2\pi\alpha t} \right\rangle_t^* \hfill (10)$

But (10) is just the conjugated conjugate CAF for the conjugated signal $x^*(t)$,

$\displaystyle R_{x^*}^{-\alpha}(\tau) = \left[ R_{(x^*)^*}^\alpha (\tau) \right]^*, \hfill (11)$

or

$\displaystyle R_{x^*}^\alpha (\tau) = \left[ R_{(x^*)^*}^{-\alpha} (\tau) \right]^*. \hfill (12)$

The symmetry relation (12) is validated by estimating the conjugate CAF of $x(t)$ for $\alpha = 2f_c$ and the conjugate CAF of $x^*(t)$ for $\alpha = -2f_c$ and plotting the results. Both functions should be even functions of $\tau$, and they should be conjugates of each other. This is shown in Figure 4.

### The Non-Conjugate Cyclic Cross Correlation Function

Let’s stay with the time-domain parameters and introduce cross functions, starting with the non-conjugate cyclic cross correlation. The basic definition is

$\displaystyle R_{xy}^\alpha(\tau) = \left\langle x(t+\tau/2) y^*(t-\tau/2) e^{-i2\pi\alpha t} \right\rangle_t \hfill (13)$

Consider the function $R_{xy}^\alpha(-\tau)$,

$\displaystyle R_{xy}^\alpha(-\tau) = \left\langle x(t-\tau/2)y^*(t+\tau/2) e^{-i2\pi\alpha t} \right\rangle_t \hfill (14)$

$\displaystyle = \left\langle [y(t+\tau/2)x^*(t-\tau/2)]^* e^{-i2\pi\alpha t} \right\rangle_t \hfill (15)$

$\displaystyle = \left[ \left\langle y(t+\tau/2)x^*(t-\tau/2) e^{-i2\pi(-\alpha)t} \right\rangle_t \right]^* \hfill (16)$

$\displaystyle = R_{yx}^{-\alpha}(\tau)^* \hfill (17)$

The relation (17) also contains the symmetry relation for cycle frequency $\alpha$ and the symmetry in ‘function order,’ that is, the symmetry relating to switching the order of $x(t)$ and $y(t)$ in the integrand. The symmetry relation for the cross cyclic CAF is illustrated for $\tau$ in Figure 5, which shows the cross correlation symmetry because $\alpha = 0$. Here $x(t)$ is the same rectangular-pulse BPSK signal as we’ve used in the previous illustrations, and $y(t)$ is the same signal shifted by $-6$ samples and subject to a different carrier phase and independent Gaussian noise. We therefore expect the cross correlation to peak at $\tau = -6$ in the XY cross function and at $\tau = 6$ in the YX version.

A more complicated example is shown in Figure 6, which features the cycle frequencies of $\pm f_{bit}$.

### The Conjugate Cyclic Cross Correlation Function

A similar analysis yields the symmetry relations for the conjugate cyclic cross correlation function,

$\displaystyle R_{xy^*}^\alpha(-\tau) = R_{yx^*}^\alpha(\tau) \hfill (18)$

and

$\displaystyle R_{yx^*}^\alpha(\tau) = \left[ R_{(x^*y^*)^*}^{-\alpha} (\tau) \right]^* \hfill (19)$

The symmetry in $\tau$ is validated in Figure 7, whereas the symmetry in cycle frequency is validated in Figure 8.

### The Non-Conjugate Spectral Correlation Function

The symmetry derivations and illustrations for the cyclic correlation functions were fun, yes, but let’s move on to the important function: spectral correlation. First, our oldest and dearest friend, the non-conjugate spectral correlation function.

The SCF is the temporal correlation between the time-series obtained from two narrowband spectral components of a signal. The narrowband components can be obtained using a simple sliding Fourier transform,

$\displaystyle X_T(t, f) = \int_{t-T/2}^{t+T/2} x(u) e^{-i2\pi f u} \, du \hfill (20)$

Their idealized correlation is

$\displaystyle \lim_{T\rightarrow\infty} \left\langle \frac{1}{T} X_T(t, f+\alpha/2) X_T^*(t, f-\alpha/2) \right\rangle_t \hfill (21)$

where we first average over all time, and then let the bandwidth $1/T$ of the narrowband components approach zero. As we go along looking at the symmetry properties of the non-conjugate, conjugate, cross, and conjugate cross SCFs, we’ll need an expression for the narrowband component of the conjugate of $x(t)$,

$\displaystyle X_{T,*}(t,f) = \int_{t-T/2}^{t+T/2} x^*(u) e^{-i2\pi f u} \, du \hfill (22)$

$\displaystyle = \left[ \int_{t-T/2}^{t+T/2} x(u) e^{-i2\pi (-f) u} \, du \right]^* \hfill (23)$

$\displaystyle = X_T(t, -f)^*. \hfill (24)$

Let’s look at the spectral correlation for the conjugated signal,

$\displaystyle S_{(x^*)}^\alpha(f) = \lim_{T\rightarrow\infty} \left\langle \frac{1}{T} X_{T,*}(t, f+\alpha/2) X_{T,*}^*(t, f-\alpha/2) \right\rangle_t \hfill (25)$

$\displaystyle = \left\langle \frac{1}{T} X_T^*(t, -(f+\alpha/2)) [ X_T^*(t, -(f-\alpha/2)) ]^* \right\rangle \hfill (26)$

$\displaystyle = \left\langle \frac{1}{T} X_T(t, -(f)+\alpha/2) X_T^*(t, (-f)+\alpha/2) \right\rangle \hfill (27)$

$\displaystyle = S_x^\alpha(-f) \hfill (28)$

So the non-conjugate SCF doesn’t possess any symmetry in $f$ on its own, but if we include the conjugated-signal SCF, we obtain a sort of symmetry.

Symmetry in cycle frequency $\alpha$ is a bit easier,

$\displaystyle S_x^{-\alpha} (f) = \lim_{T\rightarrow\infty} \left\langle \frac{1}{T} X_T(t, f-\alpha/2) X_T^*(t, f+\alpha/2) \right\rangle_t \hfill (29)$

$\displaystyle = \lim_{T\rightarrow\infty} \left[ \left\langle \frac{1}{T} X_T(t, f+\alpha/2) X_T^*(t, f-\alpha/2) \right\rangle_t \right]^* \hfill (30)$

$\displaystyle = \left[ S_x^\alpha(f) \right]^*. \hfill (31)$

So the information in the non-conjugate SCFs for negative cycle frequencies is redundant with that for positive cycle frequencies. The two forms of symmetry for the non-conjugate SCF are illustrated in Figures 9 and 10.

### The Conjugate Spectral Correlation Function

Using an analysis approach similar to that for the non-conjugate SCF, we obtain the symmetry relations for the conjugate SCF:

$\displaystyle S_{x^*}^\alpha(-f) = S_{x^*}^\alpha(f) \hfill (32)$

$\displaystyle S_{x^*}^{-\alpha}(f) = S_{(x^*)^*}^\alpha(f)^* \hfill (33)$

The conjugate SCF is even in $\tau$, and has no cycle-frequency symmetry itself, but possesses a symmetry relation with the conjugate SCF for the conjugated signal. Figures 12-14 illustrate and validate the symmetries.

### The Non-Conjugate Cross Spectral Correlation Function

The non-conjugate cross SCF is defined by

$\displaystyle S_{xy}^\alpha(f) = \lim_{T\rightarrow\infty} \left\langle \frac{1}{T} X_T(t, f+\alpha/2) Y_T^*(t, f-\alpha/2) \right\rangle_t \hfill (34)$

Using the same kinds of analysis as above, we obtain a symmetry relation for frequency $f$ of

$\displaystyle S_{xy}^\alpha(f) = S_{(x^*y^*)^*}^\alpha(-f), \hfill (35)$

and for cycle frequency $\alpha$ of

$\displaystyle S_{xy}^\alpha(f) = S_{yx}^{-\alpha}(f)^*. \hfill (36)$

Equation (36) also provides the symmetry in function order. So there is no symmetry in $f$ for the function itself, but there is between the function and the non-conjugate cross SCF for conjugated $x(t)$ and $y(t)$. Similarly, there is no symmetry in $\alpha$ for the function itself, but there is between (34) and the conjugated function-reversed version of (34) with negated cycle frequency.

The symmetries are illustrated in Figures 15 and 16.

### The Conjugate Cross Spectral Correlation Function

If you can bear it, there is one more case. The conjugate cross spectral correlation function is defined by

$\displaystyle S_{xy^*}^\alpha(f) = \lim_{T\rightarrow\infty} \left\langle \frac{1}{T} X_T(t, f+\alpha/2) Y_T(t, \alpha/2-f) \right\rangle_t \hfill (37)$

The symmetry relations are actually relatively easy to derive here. For frequency $f$ and function order we have

$\displaystyle S_{xy^*}^\alpha(f) = S_{yx^*}^\alpha(-f) \hfill (38)$

and for cycle frequency we have

$\displaystyle S_{xy^*}^\alpha(f) = \left[ S_{(y^*x^*)^*}^{-\alpha}(f) \right]^* \hfill (39)$

Once again, no simple symmetry involving just the function itself (that is, (37)); all symmetry relations depend on reversing the function order and/or involve the conjugate cross spectral correlation function for conjugated inputs. The relations (38) and (39) are illustrated and validated in Figures 17 and 18.

### Symmetries for the Spectral Coherence Function

I can hear you saying, “Yeah, great, but what about the coherence function?” And to be honest, that makes me tired, as this has been a long post. But I’m here for you. The non-conjugate spectral coherence function, greatly useful in blind CSP, is defined by

$\displaystyle C_x^\alpha(f) = \frac{S_x^\alpha(f)}{\left[ S_x^0(f+\alpha/2) S_x^0(f-\alpha/2) \right]^{1/2}} \hfill (40)$

The denominator of (40) is symmetric in $\alpha$: it is the same when $\alpha$ is replaced by $-\alpha$. But it is not symmetric in $f$. Recall that the numerator is not symmetric in $f$, but obeys the cycle-frequency symmetry

$\displaystyle S_x^{-\alpha}(f) = S_x^\alpha(f)^*$.

So we have for the non-conjugate coherence

$\displaystyle C_x^{-\alpha}(f) = C_x^{\alpha}(f)^*.\hfill (41)$

For the conjugate coherence,

$\displaystyle C_{x^*}^\alpha(f) = \frac{S_{x^*}^\alpha(f)}{\left[ S_x^0(f+\alpha/2) S_x^0(\alpha/2-f) \right]^{1/2}} \hfill (42)$

both numerator and denominator are symmetric in $f$, but not in $\alpha$. So we have the symmetry

$\displaystyle C_{x^*}^\alpha(-f) = C_{x^*}^\alpha(f) \hfill (43)$

The cross coherences don’t seem to possess much symmetry. I’ll leave those as an exercise for the reader.

### Discussion

I think the main point for CSP practitioners is that the non-conjugate spectral correlation for $\alpha \ge 0$ is redundant with respect to that for $\alpha \le 0$. If you’ve estimated the non-conjugate SCF for $\alpha \ge 0$, you’ve got all the non-conjugate information you need. On the other hand, you have to estimate the conjugate spectral correlation (or coherence) for the entire range of cycle frequency to ensure you find all the valid cycle frequencies.

Comments, corrections, suggestions, and compliments welcome below!