Inverse Z-transform

Fourier analysis is widely used in mathematics, physics, and engineering as a Fourier integral transformation pair:

$$\begin{eqnarray} B(\omega)&=&\int^{+\infty}_{-\infty} b(t)\, e^{i\omega t}\, dt ... ...)&=& \int^{+\infty}_{-\infty} B(\omega)\, e^{-i\omega t}\, d\omega\end{eqnarray}$$ (30) (31)

These integrals correspond to the sums we are working with here except for some minor details. Books in electrical engineering redefine $e^{i\omega t }$as $e^{-i\omega t }$.That is like switching $\omega$ to $-\omega$.Instead, we have chosen the sign convention of physics, which is better for wave-propagation studies (as explained in IEI). The infinite limits on the integrals result from expressing the Nyquist frequency in radians/second as $\pi/\Delta t$.Thus, as $\Delta t$ tends to zero, the Fourier sum tends to the integral. It can be shown that if a scaling divisor of $2\pi$ is introduced into either (30) or (31), then b(t) will equal $\bar b (t)$.

EXERCISES:

1. Let B(Z) = 1 + Z + Z² + Z³ + Z⁴. Graph the coefficients of B(Z) as a function of the powers of Z. Graph the coefficients of $\left[ B(Z) \right]^2$.
2. As $\omega$ moves from zero to positive frequencies, where is Z and which way does it rotate around the unit circle, clockwise or counterclockwise?
3. Identify locations on the unit circle of the following frequencies: (1) the zero frequency, (2) the Nyquist frequency, (3) negative frequencies, and (4) a frequency sampled at 10 points per wavelength.
4. Sketch the amplitude spectrum of Figure 8 from 0 to $4\pi$.