Linear Filters, Part 1

Linear Filters

Part 1: Discrete-Time Signals

In this module, as in the Difference Equations module, the principal objects of study will be sequences, that is lists of numbers. In this module, however, the sequences will be of the form

... ,y_-3, y_-2, y_-1,y₀, y₁, y₂, y₃, ...

going on forever in both directions. These are sometimes called doubly-infinite sequences. (The sequences {y₀, y₁, y₂, y₃, ...} that you studied before may be called singly-infinite sequences.) Like a singly-infinite sequence, a doubly-infinite sequence will be abbreviated as {y_k}, this time with the understanding that the index k takes on all integer values --- positive, negative, and zero.

Let S be the set of all doubly-infinite sequences of real numbers. Explain why S is a vector space -- that is, why S is closed under addition and scalar multiplication.

One place where this sort of sequence frequently appears is in engineering. Often there is a situation where you have some sort of continuous signal (for example, an electrical signal, a radio signal, a light signal, or a sound signal) that you want to measure. One way to do this is to sample the signal at regular intervals, for example, every second.

Let's be more concrete. Suppose we want to measure the following signal:

This signal was produced by a common trigonometric function. Which one? Verify that your answer gives the same graph shown above.

We can measure the strength of the signal at every quarter-second:

This gives us a pretty good picture of what the signal looks like. We can represent our samples as a sequence using the formula

y_k = cos(k/4),

where k represents the number of quarter-seconds from "time 0".

Use your computer algebra system to verify the plot above.
Use your computer algebra system to plot a similar picture of the sequence

y_k = 2 sin(k/10).

Suppose you were told that some of the values of a certain sequence were

y_-5= 0.598472

y_-4= 0.909297

y_-3= 0.997495

y_-2= 0.841471

y_-1= 0.479426

y₀= 0

y₁= -0.479426

y₂= -0.841471

y₃= -0.997495

y₄= -0.909297

y₅= -0.598472

How might you go about guessing a formula for the sequence? (Hint: you might start by looking at a plot!) Make a guess for a formula and check your answer.

Now suppose you were told that some of the values of a certain sequence were

y_-5= -2.59311

y_-4= -2.92405

y_-3= -0.566632

y_-2= 2.31175

y_-1= 3.06472

y₀= 1

y₁= -1.98411

y₂= -3.14404

y₃= -1.41335

y₄= 1.61676

y₅= 3.16044

Do you have enough information to guess a formula for the sequence? What makes you think you do or you don't?

The sequence in Step 6 was in fact calculated from the formula

y_k = cos(k) - 3 sin(k),
which corresponds to sampling the signal below every second.

If instead we sampled the same signal every tenth of a second, we would get the sequence

y_k = cos(k/10) - 3 sin(k/10).

Plot this formula from k = -50 to k = 50 . (This makes sense because 5 seconds is 50 tenths of a second.) Explain why sampling at a higher rate makes it easier to reconstruct the original signal.