Page 140 - Fundamentals of Communications Systems

4.16 Chapter Four

the output bandpass signal will have the frequency domain representation of
2
Y c (f ) = √ [δ(f − (f c + f m ) + δ(f + (f c + f m )] (4.31)
2
The complex envelopes of the input and output signals are
x z (t) = exp[ j 2π f m t] + exp[ j 6π f m t] y z (t) = 2 exp[ j 2π f m t] (4.32)
consequently it makes sense to have

2 | f |≤ 2 f m
H z (f ) = (4.33)
0 elsewhere
and H c (f ) = H z (f − f c ) + H (− f − f c ).
∗
z
Equations (4.27) and (4.14) combined with the convolution theorem of the
Fourier transform produce an expression for the Fourier transform of y c (t)
given as
1
Y c (f ) = X c (f )H c (f ) = √ [X z (f − f c ) + X (− f − f c )][H z (f − f c )
∗
z
2
∗
+ H (− f − f c )]
z
Since both X z (f ) and H z (f ) only take nonzero values in [−B T /2, B T /2], the
cross terms in this expression will be zero and Y c (f ) is given by
1
Y c (f ) = √ [X z (f − f c )H z (f − f c ) + X (− f − f c )H (− f − f c )] (4.34)
∗
∗
z
z
2
Since y c (t) will also be a bandpass signal, it will also have a complex baseband
representation. A comparison of Eq. (4.34) with Eq. (4.14) demonstrates the
Fourier transform of the complex envelope of y c (t), y z (t), is given as
Y z (f ) = X z (f )H z (f )
Linear system theory produces the desired form
∞

y z (t) = x z (τ)h z (t − τ)dτ = x z (t) ∗ h z (t) (4.35)
−∞
EXAMPLE 4.8
(Example 4.1 cont.) The input signal and Fourier transform are
x z (t) = 2 cos(2π f m t) + j sin(2π f m t) X z (f ) = 1.5δ(f − f m ) + 0.5δ(f + f m )
Assume a bandpass filter with
⎧ f
⎪ j −f m ≤ f ≤ f m
f m
⎪
⎪
2 −2 f m ≤ f ≤ 2 f m ⎨ j f m ≤ f ≤ 2 f m
H I (f ) = H Q (f ) = (4.36)
0 elsewhere ⎪− j −2 f m ≤ f ≤−f m
⎪
⎪
0 elsewhere
⎩