Problem Set 3.1 Spring 2009
Composition of Functions and Vectors for LTI Systems Theory

1. Given the three functions f(x), g(x), and h(x) below, compute and simplify the various compositions of these functions.
   
   

   f(x) = x2 – 3x

   g(x) =2 x– 4

   h(x) = x2 + 2

   1. g(h(3))
      
      
   2. h(g(3))
      
      
   3. f(g(t))
      
      
   4. h(g(f(1)))
      
      

   5. What simple conclusion about composing functions can you draw from (a) and (b)?

2. This problem will use the data set from Section 3.2 reproduced in the table below.

   Frequency band (kHz) (Column A)	2	4	6	8	10	12	14	16	18	20
   Amplitude (Column B)	148	150	153	145	157	149	150	150	142	145

   The frequencies given in row 1 also may be represented as an array:  f = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] KHz.

   1. Assume that this is the amplitude spectrum of the source of some sound. Now suppose the sound is generated in a room with the following transfer function:

      r(f) = [1.3, 1.2, 1.1, 0.8, 0.8, 0.5, 0.6, 1.0, 1.0, 1.0]

      Next assume that the sound in the room is picked up by a microphone and recorded. The transfer function of the microphone is

      m(f) = [1.0, 1.0, 1.5, 2.0, 2.0, 2.0, 1.5, 1.2, 1.0, 1.0]

      What is the amplitude spectrum of the recorded sound? Hint:  To represent this complex system in Excel, create a table or matrix of data.  For example, one might represent sound frequency, f, in one column, the sound source in a second column, the room transfer function in a third column, and so on.  To apply a transfer function to an input signal, multiply the signal amplitude by the gain specified by the transfer function at each frequency.  Recall that multiplication is used when signal levels and transfer functions are given in linear amplitude units.

   2. Before the recording is pressed into millions of CDs, a sound engineer may want to "correct" the recorded sound by filtering the recording through an equalizer. Determine the transfer function of the equalizer e(f) if it is to produce a new recording that is true to the original source.

   3. Sketch the amplitude spectra of the original source sound, each transfer function, and each intermediate result as the sound progresses through the systems, and the final spectrum after equalization (total 7 plots). 
      

3. Find the horizontal and vertical components of the following vectors.  Hint: First sketch a graph of the given information.
   1. 40 at 80°
      
      
   2. 25 at /3 radians
      
      
   3. 50 at 45°
   
   Add the following sets of vectors. Hint: First sketch a graph of the given information. Either represent as <x,y> or as length<angle.

4. 10 at 0°, 30 at 90°
   
   
5. 20 at 0°, 40 at -90°, 50 at 90°
   
   
6. 12 at /3 radians, 18 at /8 radians