Homework Set 5 for Advanced Engineering Mathematics

In this homework set we will study the so-called Discrete Fourier Transform, becauseit provides a nice review of many concepts that we have studied already.

Task 1 (Discrete-Time Signals)

So far, we have studied functions defined on an interval, in particular, 2pi-periodicfunctions defined on (,) or their prototypes defined on an interval of length2pi, e.g., on [0, 2pi] or on [pi, pi]. These functions may correspond to the result ofmeasuring some quantity for every moment in time. In this context, such a function iscalled a continuous-time signal, or CT signal for short.

But in engineering we often measure a quantity only at discrete moments in time. Forinstance, lets say we measured a value of 1 (arbitrary unit) at time 0 (seconds), a valueof 4 at time 1, a value of 2 at time 2, and a value of 3 at time 3

Draw the graph of the function

f(n) =

1 if n = 04 if n = 12 if n = 23 if n = 3

To stress that the measurements were taken at discrete times, people often use variablenames k or n instead of t or x. (Hint: Use vertical lines similarly to our visualization ofa spectrum of a periodic function. A spectrum is a sequence of amplitudes measuredat discrete frequencies.)

Now extend this graph periodically with a period of 4, i.e., the above four values formthe prototype of a 4-periodic function. For example, at n = 4 the value is 1 and atn = 1 the value is 3. Draw this 4-periodic function for 6 n 10.A function is 4-periodic, if

f(n) = f(n+ 4) for all n.

This periodic discrete-time signal can be represented by the infinite sequence

(. . . , x1, x0, x1, . . .) = (. . . , 3, 1, 4, . . .)

while its prototype is simply represented by a vector

(1, 4, 2, 3).

Note, that I chose the first vector component to correspond to the index 0. This is mychoice, like I could choose [0, 2pi] or [pi, pi] or any other interval of length 2pi.

Task 2 (Linear Operation: Time Shift)

The complex exponentials were good basis functions for continuous-time functionsbecause they were eigenfunctions for the important linear operations differentiation,integration, and time shift.

Differentiation and integration do not make sense for discrete-time signals. Why?(Think of the limit processes that are used to define differentiation and integrationrespectively.) But a time shift is still a sensible operation:

The sequence (xn) is shifted to T (xn) = (xn1).

Take (xn) = (. . . , x1, x0, x1, . . .) = (. . . , 3, 1, 4, . . .) from the previous example. Thesecond term of the shifted sequence T (xk) is x21 = x1 = 4. Draw the graph of theshifted 4-periodic function. Convince yourself that it is the graph from task 1 shiftedby 1 unit to the right.

The sequences (. . . , x1, x0, x1, . . .) form a vector space. How are vector addition andscalar multiplication performed?

Show that this time-shift operation is a linear map from the vector space of sequencesto itself.

Now, consider this time shift only on the prototype instead of on the 4-periodic se-quence. If we start with the prototype (1, 4, 2, 3), what will the shifted prototype be?(Hint: If you cannot find a solution, look at the next task.)

Task 3 (Matrix of a Linear Map)

The (periodic) shift

T

x0x1x2x3

=x3x0x1x2

is a linear map from the vector space of column vectors with four components to itself.Find its matrix representation with respect to the standard basis. In other words,compute the matrix (tij), so that

x3x0x1x2

= t11 t14... ...t41 t44

x0x1x2x3

Task 4 (Eigenvalue Problem)

Find the (complex) eigenvalues and eigenvectors (choose first component 1) of thematrix

T =

0 0 0 11 0 0 00 1 0 00 0 1 0

(Hint: Laplace expansion is useful to compute the determinant for the characteristicequation.) Draw the eigenvalues as points in the complex number plane. Where arethey located with respect to the unit circle? What is the relationship with the complexexponential eix?

Hint: It may be faster to compute the eigenvectors by looking at the eigenvalue equationx3x0x1x2

= x0x1x2x3

starting with x0 = 1.

Show that these eigenvectors form an orthogonal basis of the vector space of columnvectors with four (complex) components. Recall that the inner product uses complexconjugation.

Task 5 (Discrete Complex Exponentials)

Find the prototypes (x0, x1, x2, x3) of the 4-periodic discrete-time complex exponentialsxn = e

ink2pi/4 for k = 0, 1, . . . , 4 (i.e., there are five prototypes to consider).

Note, n is the index, i.e., corresponds to the variable x in the continous-time complexexponentials eikx. The frequencies are k2pi/4. This fits our discussion of 2L-periodicfunctions. Here, 2L = 4.

Compare your results with the eigenvectors from the preceding task.

Task 5 (Coordinates)

Find the coordinates of (1, 4, 2, 3) with respect to the so-called discrete Fourier Basisfrom the previous task. (These coordinates are the Fourier coefficients.)

Use an inverse matrix to do this. How does orthogonality help in finding the inver-se matrix? Since the matrix has complex entries, you will need to transpose and toconjugate. Since the vectors are not unit vectors, you also need a scaling factor.

Challenge (Totally Voluntary - only for students for whom all this is easy)

What is the Fourier transform of a non-periodic discrete-time signal?