2.1  To Learn in This Chapter

The skills we aim to develop in this chapter are:

• Apply basic concepts of linear algebra such as inner products and projections, and then make analogies between vectors and signals to better understand transforms
• Use inner products to efficiently obtain the transform coefficients when the basis functions are orthogonal
• Get familiar with the most used block transforms: DCT and DFT
• Observe how wavelets localize information in both frequency and time by using basis functions with limited time support
• Interpret transforms (Fourier, Z, Laplace) as obtaining coefficients given by the inner product between the signal to be transformed and the corresponding basis function
• Know what Hermitian symmetry is and when it happens in the signal spectrum
• Understand advantages of linear transformations such as DCT for coding applications
• Understand the relations among the four pairs of equations for Fourier analysis with eternal (infinite duration) sinusoids as basis functions: Fourier series (FS), Fourier transform (FT) and their discrete-time (DT) counterparts: DTFS and DTFT
• Understand how to use an efficient (fast) FFT algorithm to compute the block transform DFT
• Learn how the $N$-points FFT with basis functions that last $N$ samples can be used to estimate the FS, FT, DTFS and DTFT
• Note the three most used alternatives of normalization factors for the FFT and learn which one to use according to the application: unitary, DFT as samples of the DTFT and DFT coinciding with the DTFS
• Properly interpret the results of DFT and DTFS in spite of their equations differing simply by a constant factor $N$
• Observe that the Fourier transform can be expressed in terms of linear (Hz) or angular (rad/s) frequencies, and how this impacts the spectra $X(f)$ and $X(\omega )$ when impulses are involved
• Learn the relation $X(e^{\mathrm{\Omega }})=F_{s}X(\omega )$ between the discrete-time spectrum $X(e^{\mathrm{\Omega }})$ and its continuous-time version $X(\omega )$ by using $\omega =\mathrm{\Omega }{F}_s$
• Relate the continuous-time Fourier transform $X(\omega )$ and Laplace transform $X(s)$ based on their respective basis functions, noting that $e^{\mathit{j\omega t}}$ is the special case of $e^{\mathit{st}}$ when the complex-valued independent variable $s$ is $s=\mathit{j\omega }$
• Relate the discrete-time Fourier transform (DTFT) $X(e^{j\mathrm{\Omega }})$ and Z transform $X(z)$ based on their respective basis functions, noting that $e^{j\mathrm{\Omega }n}$ is the special case of $z^n$ when the complex-valued independent variable $z$ is $z=e^{j\mathrm{\Omega }}$
• Calculate and properly interpret the region of convergence (ROC) for the Laplace and Z transforms
• Obtain the inverse Laplace and Z transforms using partial fraction expansion, even when there are poles with multiplicity larger than one
• Observe the notation $X(e^{j\mathrm{\Omega }})$ aims at making explicit that $\mathrm{\Omega }$ is an angle (while in continuous-time, the unit of $\omega$ is rad/s) and, consequently, $X(e^{j\mathrm{\Omega }})$ is periodic with fundamental period $\mathrm{\Omega }=2\pi$
• Obtain by inspection the Fourier series coefficients of periodic signals composed by harmonic sinusoids

Transforms are a very important tool in several applications. The continuous-time Fourier transform, for example, provides an alternative “view” $X(f)$ of a signal $x(t)$. Sometimes this extra view is essential for efficiently solving a problem. For instance, designing a filter to eliminate a frequency band is easier in the transform domain. Few examples of transforms and applications can be found in Table 2.1.

We start our study focusing on linear transforms and making a convenient connection with linear algebra.