4.1  To Learn in This Chapter

The skills we aim to develop in this chapter are:

• Use finite-duration windows to model the extraction of a segment from a signal with a potentially infinite duration
• Apply windows to better control resolution and leakage in spectral analysis
• Interpret the power spectral density (PSD)
• Distinguish PSD, ESD and the mean-square (power) spectrum (MS spectrum), knowing the cases in which one should be adopted
• Estimate PSD, ESD and the mean-square (power) spectrum (MS spectrum) using the FFT, and be familiar with concepts such as the periodogram and Welch’s method
• Recognize that the periodogram variance does not decrease with the number of samples, and understand how Welch’s method decreases the variance
• Learn how the noise floor decreases with an increased number $N_{\mathrm{\text{fft}}}$ of FFT points
• Estimate the PSD from the autocorrelation
• Distinguish the unilateral and bilateral representations
• Perform spectrum analysis with a computer
• Be able to mathematically model the spectrum leakage when windowing a signal via the circular convolution of the original spectrum with the window’s spectrum
• Observe that the FFT resolution $\mathrm{\Delta }f=F_s∕N_{\mathrm{\text{fft}}}$ improves as $N_{\mathrm{\text{fft}}}$ gets larger, but this cannot recover the leakage that occurred due to windowing
• Know the Z transform property to easily normalize the samples of an impulse response $h[n]$ by their summation to have a gain of 0 dB at the DC $H(e^{j\mathrm{\Omega }}){|}_{\mathrm{\Omega }=0}$
• Understand the picket fence effect when using FFTs for spectral analysis
• Learn the reasons that lead a stronger sinusoid that is not bin-centered to completely hide a weaker sinusoid in FFT-based spectral analysis
• Estimate not only the sinusoid frequency but also its correct amplitude and correct the window scalloping loss
• Distinguish a spectrum analyzer and a vector network analyzer (VNA)
• Obtain the output PSD when the input to a LTI system is a WSS process with a given input PSD
• Model the filtering of white noise through LTI systems
• Perform parametric PSD estimation via autoregressive (AR) models