Frequently Asked Questions
Question

• How do I test to see if something is linear? (Exam Practice problem 8)
  
  Answer: In order to test for linearity, ask whether superposition applies:
  Suppose you had a pair of waveforms x[.] and y[.] (i.e., with respective values x[n] and y[n] at time n)
  that satisfied this difference equation, i.e.,
  
  y[n]=alpha.x[n]+beta.x[n-1]
  
  and another pair x'[.] and y'[.] that satisfied it, i.e., such that
  
  y'[n]=alpha.x'[n]+beta.x'[n-1]
  
  would the pair
  
  wx[.]+w'x'[.] and wy[.]+w'y'[.]
  
  for arbitrary scalar weights w and w' also satisfy the definining equation? i.e., would it be true that
  
  wy[n]+w'y'[n] = alpha.(wx[n]+w'x'[n]) + beta.(wx[n-1]+w'x'[n-1])
  
  And the answer would be YES, as you can see by adding w times the first equation above to w' times the second equation above.
  
  Similarly for time invariance: If x[.] and y[.] satisfy the given equation, is it true that x'[.] and y'[.] will satisfy it,
  where x'[n]=x[n-s] and y'[n]=y[n-s], and s is an arbitrary (integer) time-shift. And the answer is again YES,
  since the given equation holds for all n, and in particular if you replace the n in the equation by n-s.

Question

• How do we know if inter-symbol interference (ISI) is carrying over
  from the previous bits? (exam practice problem 10)
  
  Answer: since the unit sample response takes non-zero values over an interval of 9 samples
  
  (1,2,...,9) long, the value of the output at a given sample is determined by the values of 9 consecutive input samples,
  which will depend on the values of 2 or 3 input bits (for example, if b[] is the input bit sequence, and x[] is the transmitted
  samples sequence, then the first bit b[0] is transmitted as x[0]=x[1]=...=x[4]=b[0], the second bit b[1] is transmitted as
  
  x[5]=x[6]=...=x[9]=b[1], etc.
  (b[k] transmitted as x[5k]=x[5k+1]=...=x[5k+4]=b[k] for k=0,1,2,...)
  hence the output bit x[9] depends on (x[0],x[1],...,x[9]), i.e. on b[0] and b[1], while x[10] depends on (x[1],x[2],...,x[10]),
  
  i.e. on b[0],b[1], and b[3]. The fact that some output samples depend on three input bits
  (instead on a single input bit, as would be in a memory-less channel with unit sample response h[n]=delta[n]) can be interpreted as
  "the inter-symbol interference [...] carrying over from the two previous transmitted bits'"
  
  

Question

• (Exam practice problem 11) - Where does the left side of the equation
  ( 0.5 ) * H( 0 ) * ( 1 + 0.5 ) = 1 come from? How do I use the given picture
  for this? Where did the picture come from in 11c? For 11d, where did the
  calculations come from?
  
  Answer: For 11a, the area under the PDF is a triangle of height H(0) and
  base length of 1 + 0.5. The integral of a triangle is 1/2 * Base * Height
  
  For 11c, the picture shows 3 plots, the PDF of the received voltage, and two
  auxillary graphs which were used to compute the magenta plot:
  
  blue (partially hidden by magenta)
  
  the pdf of recieved voltage when -0.5 goes in,
  times the probability (3/4) of sending -0.5 in red(also partially hidden):
  the pdf of received voltage when +0.5 goes in,
  times the probability (1/4) of sending +0.5 in
  
  The magenta plot is the SUM of the two.
  
  
  To be honest, the magenta plot should not be used at all: the error is the
  area under the blue line from the threshold (0 in this case) to +infinity
  plus the area under the red line from -infinity up to the threshold
  (the area is zero, since the red line is zero before 0)
  
  For 11d, you will have to calculate the value of BER (ie error probability)
  for an arbitrary value of the threshold, and then to minimize the result
  over all possible thresholds (-infinity to +infinity)