(a) Let S denote an incrementally linear system, and let x1 [n] be an arbitrary input si...

(a) Let S denote an incrementally linear system, and let x₁ [n] be an arbitrary input signal to S with corresponding output y₁ [n]. Consider the system illustrated in Figure P1 .47(a). Show that this system is linear and that, in fact, the overall input-output relationship between x[n] and y[n] does not depend on the particular choice of x₁ [n].

(b) Use the result of part (a) to show that S can be represented in the form shown in Figure 1.48.

(c) Which of the following systems are incrementally linear? Justify your answers, and if a system is incrementally linear, identify the linear system L and the zero-input response y₀ [n] or y₀ (t) for the representation of the system as shown in the Figuree 1.48.

(iv) The system depicted in Figure P1.47(b).

(v) The system depicted in Figure P1.47(c).

(d) Suppose that a particular incrementally linear system has a representation as in Figure 1.48, with L denoting the linear system and y₀ [n] the zero-input response. Show that S is time invariant if and only if L is a time-invariant system and y₀ [n] is constant.

The complex number z can be expressed in several ways. The Cartesian or rectangular form for z is

where j = √-1 and x and y are real numbers referred to respectively as the real part and the imaginary part of z. As we indicated earlier, we will often use the notation

The complex number z can also be represented in polar form as

where r > 0 is the magnitude of z and θ is the angle or phase of z. These quantities will often be written as

The relationship between these two representations of complex numbers can be determined either from Euler's relation,

or by plotting z in the complex plane, as shown in Figure P1.48, in which the coordinate axes are along the vertical axis. With respect to this graphical representation, x and y are the Cartesian coordinates of z, and r and θ are its polar coordinates.