PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Time Domain Representation of Continuous and Discrete Signals 23

R.1.68 Any real function or sequence can be expressed as a sum of its even part ( fe) plus its

odd part ( fo) as indicated by the following equation:

ƒ(t) = ƒe(t) + ƒo(t)

where ƒe(t) = 1/2 [ƒ(t) + ƒ(−t)] and ƒo(t) = 1/2 [ƒ(t) − ƒ(−t)] for the analog case, and

ƒ(n) = ƒe(n) + ƒo(n), where ƒe(n) = 1/2 [ƒ(n) + ƒ(−n)] and ƒo (t) = 1/2 [ƒ (n) − ƒ (−n)] for

the discrete case, assuming the sequences are real.

R.1.69 The average value of the function f(t) in the interval −T/2 ≤ t ≤ T/2 is given by

f

T

ave ftdt

T T

T


lim ( )

/

/

→∞ ∫

















1

2

2



Observe that the average value fave is contained in the even portion of f(t) { fe(t)},

since the contribution of the odd portion is always zero.

R.1.70 The general algebraic rules governing even and odd symmetric functions are sum-

marized as follows:

a. The sum of two even functions is also even.

b. The product of two even functions is also even.

c. The product of two odd functions is even.

d. An even function squared becomes even.

e. An odd function squared becomes even.

f. The sum of two odd functions is also odd.

g. The sum of an even plus an odd function is neither even nor odd.

h. The product of an even by an odd function is odd.

R.1.71 Any analog signal f(t), or discrete sequence f(n), of the independent variables either

t or n, can be transformed with respect to the independent variable (t or n) in the

following ways:

a. Time transformation

i. Reversal or refl ection returns f(−t) or f(−n).

ii. Time scaling by a returns f(at) or f(an) {expansion (a < 1 ), or compression

(a > 1 )}.

iii. Time shifting by to returns f(t − t 0 ) or f(n − n 0 ). If to > 0 then f(t) is shifted to

the right by to, and if to < 0 then f(t) is shifted to the left by to.

b. Amplitude transformation

i. Inversion returns −f(t) or −f(n).

ii. Amplifi cation or attenuation by A returns A f(t) or A f(n). If A > 1 , which means

amplifi cation and if A < 1 , which means attenuation.

iii. Direct current (DC) shifting by A returns A + f(t) or A + f(n). If A > 0 , which

means f(t) moves up by A and if A < 0 , which means f(t) moves down by A.

R.1.72 Recall that given a continuous time signal f(t), the signal f(t − t 1 ) is the signal f(t)

shifted t 1 units to the right, and f(t + t 2 ) represents the signal f(t) shifted t 2 units to

the left, where t 1 and t 2 are positive, real numbers.

R.1.73 For example, let f(t) be the function shown in Figure 1.21.

Sketch the functions f(t − 1), f(t − 2), f(t + 1), and f(t + 2).