Generation of 1000 Samples of Zero-Mean Unit-Variance Gaussian White Noise

In this problem, we have a set of 1000 samples generated from a Gaussian white noise process with zero mean and unit variance. Our objective is to estimate the autocorrelation sequence of this random process. The autocorrelation sequence reveals how a signal correlates with itself at different time lags, which is crucial for various signal processing applications such as filtering, prediction, and system identification. To implement this in code, we can first generate the noise samples using a random number generator. In MATLAB, this can be achieved with the command: `noise = randn(1, 1000);` which produces normally distributed random numbers with zero mean and unit variance. For other programming languages, similar functions like `numpy.random.randn()` in Python can be used. The autocorrelation function measures the similarity between a signal and its time-shifted version. We can estimate it using methods like the direct computation approach or built-in functions. In MATLAB, the `xcorr()` function can be employed: `autocorr_seq = xcorr(noise, 'biased');` where the 'biased' option provides a biased estimate suitable for finite-length sequences. Alternatively, we can implement it manually by calculating the sum of products for each lag: `for lag = 0:N-1; autocorr(lag+1) = sum(noise(1:N-lag) .* noise(lag+1:N)); end` This estimation helps us understand the signal's characteristics, such as periodicity and noise properties, where white noise should theoretically show near-zero autocorrelation for all non-zero lags.