sm10_17

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Problem 10.17. For a rectangular pulse shape, by how much does null-to-null 
transmission bandwidth increase, if the transmission rate is increased by a factor of three? 
 
Solution 
Without loss of generality, consider the baseband BPSK signal: 
 
,)()( ∑ −=
k
k kTthbts 
 
where T is the symbol duration, bk = +1 or -1 for transmitted 1 or 0, respectively. The 
pulse h(t) is rectangular, 
 
⎟⎠
⎞⎜⎝
⎛ −=
T
Ttth 2/rect)( . 
 
The Fourier transform H(f) of h(t) is 
 
2 / 2( ) sinc( )
sin( )
j fT
j fT
H f T fT e
fTT e
fT
π
ππ
π
−
−
= ⋅
= . 
 
Inspecting a plot of the sinc function, we see the null-to-null transmission bandwidth of 
H(f) is B = 2/T. When the transmission rate is increased by a factor three, we have the 
new symbol duration . The null-to-null bandwidth / 3T T′ = 2 / 3B T B′ ′= = , increased by 
a factor of 3. 
 
 
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