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The discrete-time Fourier transform is a linear operator. That is to say, if X 1 (ejw)is the DTFT of xl(n), and X2(eJw) the DTFT of x2(n), is

Shifting a sequence in time results in the multiplication of the DTFT by a complex exponential (linear phase term): x(n --no)

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Node concat(Node list1, Node list2) // returns: a new list that contains a copy of list1, followed by // a copy of list2; For example, if list1 is {22, 33, 44, 55} and list2 is {66, 77, 88, 99}, then concat(list1, list2) will return the new list {22, 33, 44, 55, 44, 55, 66, 77, 88}. Note

3.10 Write and test this method:

e - j n o w x (e j w )

Time-reversing a sequence results in a frequency reversal of the DTFT :

that the three lists should be completely independent of each other. Changing one list should have no effect upon the others. 3.9 Write and test this method:

int get(Node list, int i) // returns the value of element number i;

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Multiplying a sequence by a complex exponential results in a shift in frequency of the DTFT : ejnwx(n) E x(,j(w-w)) ! J Thus, modulating a sequence by a cosine of frequency

the spectrum up and down in frequency by oo:

void set(Node list, int i, int x) // replaces the value of element number i with x; For example, if list is {22, 33, 44, 55, 66, 77, 88, 99}, then set(list, 2, 50) will change list to {22, 33, 50, 55, 66, 44, 88, 99}.

For example, if list is {22, 33, 44, 55, 66, 77, 88, 99}, then get(list, 2) will return 44. 3.11 Write and test this method:

Perhaps the most important result in linear systems theory is that convolution in the time domain is equivalent to multiplication in the frequency domain. Specifically, this theorem says that the DTFT of a sequence that is formed by convolving two sequences, x(n) and h(n), is the product of the DTFTs of x(n) and h(n):

As with the time-shift and modulation properties, there is a dual to the convolution theorem that states that multiplication in the time domain corresponds to (periodic) convolution in the frequency domain:

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3.10 Write and test this method:

void put(Node list, int i, int x) // inserts x as element number i;

Parseval's theorem is referred to as the conservation of energy theorem, because it states that the DTFT operator preserves energy when going from the time domain into the frequency domain.

int get(Node list, int i) // returns the value of element number i;

In this section, we present some applications of the DTFT in discrete-time signal analysis. These include finding the frequency response of an LSI system that is described by a difference equation, performing convolutions, solving difference equations that have zero initial conditions, and designing inverse systems.

For example, if list is {22, 33, 44, 55, 66, 77, 88, 99}, then put(list, 3, 50) will change list to {22, 33, 44, 50, 55, 66, 44, 88, 99}. Hint: if i = 0, replace the value of the first node with x, and insert a new node immediately after it that contains the previous fist value. 3.12 Write and test this method:

An important subclass of LSI systems contains those whose input, x(n), and output, y(n), are related by a linear constant coefficient difference equation (LCCDE):

For example, if list is {22, 33, 44, 55, 66, 77, 88, 99}, then get(list, 2) will return 44. 3.11 Write and test this method:

void swap(Node list, int i, int j) // swaps the ith element with the jth element;

The linearity and shift properties of the DTFT may be used to express this difference equation in the frequency domain as follows: y(eJw)= -

a(k)e-jkwy(ejw)

void put(Node list, int i, int x) // inserts x as element number i;

For example, if list is {22, 33, 44, 55, 66, 77, 88, 99}, then swap(list, 2, 5) will change list to {22, 33, 77, 55, 66, 44, 88, 99}. 3.13 Write and test this method:

b(k)e-~*~x(e~~)

EXAMPLE 2.7.1 Consider the linear shift-invariant system characterized by the second-order linear constant coefficient difference equation

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