EE313 Linear Systems and Signals - Homework #10 Hints

• Computing phase responses: One can compute the phase a complex number with angle in Matlab/Mathscript. The angle also works for a vector of complex values, e.g. samples of a frequency response taken at different frequencies. The angle command returns a phase between [-pi/2, pi/2], which is known as wrapped phase.

  One could compute the unwrapped phase by applying the phase command to a vector of complex values.

  Here is a simple example illustrating unwrapped phase. Consider an ideal delay. The transfer function is H[z] = z^-1. The frequency response is H[e^{j w}] = e^{-j w}. The phase is -w. If we were to plot the phase response as arctan( e^{-j w} ), then we would see jumps of discontunity at ..., -3 pi/2, -pi/2, pi/2, 3 pi/2, ... instead of a straight line.

• Problem 10.1: Realize the system means to draw the block diagram. It is a realization because components in the block diagram map directly to a hardware or software implementation.

  In the discrete-time domain, you can think of the arcs in the block diagrams as containing a number (or really a stream of numbers). The units of numbers on the arcs do not matter.

  The cascade form can be found by factoring the transfer function into a product of first-order transfer functions. The parallel form can be found by apply partial fractions decomposition on the transfer function.

  Please use the new transfer function in solving this problem.

                      -1       
                 1 + z         
  H(z) = ----------------------
                  -1         -2
         1 + 1.8 z   + 0.82 z  
  

  The new transfer function has a zero at z = -1 and poles are at -0.9 + j 0.1 and -0.9 - j 0.1.

• Problem 10.2: This is a discrete-time tapped delay line. Discrete-time tapped delay lines, also known as digital finite impulse response (FIR) filters, are commonly used in communication, audio, radar, and sonar systems. In these systems, it is more common for the coefficients of the polynomial transfer function in ascending powers of z^(-k) to be symmetric or anti-symmetric about the midpoint. In this problem, however, the coefficients are neither symmetric or anti-symmetric about the midpoint.

• Problem 10.3: Assume that the parameter a is complex-valued.

  In Matlab/Mathscript, compute the magnitude of a complex number using abs. This function also works for a vector of values.

• Problem 10.4: All-pass filters are used to compensate for phase distortion. Phase distortion can occur when a signal has been filtered by an infinite impulse response (IIR) filter. IIR filters can model the LTI effects in transmission lines as used in a wireline modem. IIR filters are used in A/D and D/A converters, so an all-pass filter can follow an A/D converter to compensate for the phase distortion introduced by the IIR filter.

  Please see handout J for the derivation of the pole-zero relationship for a first-order all-pass filter.

  Interesting questions (although not required for this problem): how would one design a third-order all-pass filer? how would one design a fourth-order all-pass filter?

• Problem 10.5: When using the arctangent function to determine the angle for the compact trigonometric series, there can be a division by zero if the cosine series coefficient is 0. In that case, the arctangent function has a value of pi/2 if bⁿ is positive or -pi/2 if bⁿ is negative. A better version of the arctangent function has two arguments where x is the horizontal component and y is the vertical component:
  • ArcTan[x, y] in Mathematica
  • atan2(y, x) in Matlab/Mathscript
  If you pass y/x as the lone argument to the arctangent function, then the arctangent function can only return an angle in the range -pi/2 ≤ angle < pi/2. The two argument version can return an angle over an entire period, -pi ≤ angle < pi.

bevans@ece.utexas.edu