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 ( f e ) plus its
odd part ( f o ) as indicated by the following equation:
ƒ(t) = ƒ e (t) + ƒ o (t)
where ƒ e (t) = 1/2 [ƒ(t) + ƒ(−t)] and ƒ o (t) = 1/2 [ƒ(t) − ƒ(−t)] for the analog case, and
ƒ(n) = ƒ e (n) + ƒ o (n), where ƒ e (n) = 1/2 [ƒ(n) + ƒ(−n)] and ƒ o (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
f t dt
ave
T
T
T
ϭ lim
( )
/
/
→∞
∫
1
2
2
Ϫ
Observe that the average value f ave is contained in the even portion of f(t) { f e (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 summarized 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 t o returns f(t − t 0 ) or f(n − n 0 ). If t o > 0 then f(t) is shifted to
the right by t o , and if t o < 0 then f(t) is shifted to the left by t o .
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).
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R.1.68 Any real function or sequence can be expressed as a sum of its even part ( f e ) plus its
odd part ( f o ) as indicated by the following equation:
ƒ(t) = ƒ e (t) + ƒ o (t)
where ƒ e (t) = 1/2 [ƒ(t) + ƒ(−t)] and ƒ o (t) = 1/2 [ƒ(t) − ƒ(−t)] for the analog case, and
ƒ(n) = ƒ e (n) + ƒ o (n), where ƒ e (n) = 1/2 [ƒ(n) + ƒ(−n)] and ƒ o (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
f t dt
ave
T
T
T
ϭ lim
( )
/
/
→∞
∫
1
2
2
Ϫ
Observe that the average value f ave is contained in the even portion of f(t) { f e (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 summarized 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 t o returns f(t − t 0 ) or f(n − n 0 ). If t o > 0 then f(t) is shifted to
the right by t o , and if t o < 0 then f(t) is shifted to the left by t o .
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).
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