# Resolution in Time, Frequency, and Cycle Frequency for CSP Estimators

In this post, we look at the ability of various CSP estimators to distinguish cycle frequencies, temporal changes in cyclostationarity, and spectral features. These abilities are quantified by the resolution properties of CSP estimators.

### Resolution Parameters in CSP: Preview

Consider performing some CSP estimation task, such as using the frequency-smoothing method, time-smoothing method, or strip spectral correlation analyzer method of estimating the spectral correlation function. The estimate employs $T$ seconds of data.

Then the temporal resolution $\Delta t$ of the estimate is approximately $T$, the cycle-frequency resolution $\Delta \alpha$ is about $1/T$, and the spectral resolution $\Delta f$ depends strongly on the particular estimator and its parameters. The resolution product $\Delta f \Delta t$ was discussed in this post. The fundamental result for the resolution product is that it must be very much larger than unity in order to obtain an SCF estimate with low variance.

# The Cyclic Autocorrelation for Rectangular-Pulse BPSK

The cyclic autocorrelation for rectangular-pulse BPSK can be derived as a relatively simple closed-form expression (see My Papers [6] for example or The Literature [R1]). It can be estimated in a variety of ways, which we will discuss in future posts. The non-conjugate cycle frequencies for the signal are harmonics of the bit rate, $k f_{bit}$, and the conjugate cycle frequencies are the non-conjugate cycle frequencies offset by the doubled carrier, or $2f_c + k f_{bit}$.

Recall that the simulated rectangular-pulse BPSK signal has $10$ samples per bit, or a bit rate of $0.1$, and a carrier offset of $0.05$, all in normalized units (meaning the sampling rate is unity). We’ve previously selected a sampling rate of $1.0$ MHz to provide a little physical realism. This means the bit rate is $100$ kHz and the carrier offset frequency is $50$ kHz. From these numbers, we see that the non-conjugate cycle frequencies are $k 100$ kHz, and that the conjugate cycle frequencies are $2(50) + k 100$ kHz, or $100 + k 100$ kHz.

# The Cyclic Autocorrelation

In this post, I introduce the cyclic autocorrelation function (CAF). The easiest way to do this is to first review the conventional autocorrelation function. Suppose we have a complex-valued signal $x(t)$ defined on a suitable probability space. Then the mean value of $x(t)$ is given by

$M_x(t, \tau) = E[x(t + \tau)]. \hfill (1)$

For stationary signals, and many cyclostationary signals, this mean value is independent of the lag parameter $\tau$, so that

$\displaystyle M_x(t, \tau_1) = M_x(t, \tau_2) = M_x(t, 0) = M_x(t). \hfill (2)$

The autocorrelation function is the correlation between the random variables corresponding to two time instants of the random signal, or

$\displaystyle R_x(t_1, t_2) = E[x(t_1)x^*(t_2)]. \hfill (3)$