Digital Signal Processing Reference
In-Depth Information
the receiver knows exactly where to sample). At time t
T, the
receiver will be sampling the first symbol. Notice how the two
later symbols have zero crossings at t
¼
¼
T, and so have no
contribution at this instant. At t
0, the receiver will be sampling
the value of the second symbol. Again, the other symbols, such as
first and third adjacent symbols, have zero crossings at t
¼
0, and
have no contribution. If the bandwidth of the filter is reduced to
less than 500 kHz (R / 2) then in the frequency domain these
pulses would widen (remember that the narrower the frequency
spectrum, the longer the time response, and vice versa).
Figure 17.7
shows the result if the F
cutoff
of the pulse shaping filter
is narrowed to 250 kHz, or R / 4.
¼
F
cutoff
= R/4 = 1/4T
time axis
-4T
-2T
-T
T
2T
3T
4T
-3T
Figure 17.7. Narrow
lter impulse response.
0, the receiver will be
sampling contributions from all three pulses. At each sampling
point of t equal to
In this case, notice how at time t
¼
the signal is going to
have contributions from many nearby symbols, preventing
detection of any specific symbol. This phenomenon is known as
inter-symbol interference (ISI), and shows that transmitting
symbols at a rate R requires at least R Hz (or 1 / T Hz) in the
passband frequency spectrum. At baseband, the equivalent two-
dimensional (complex) spectrum is from
.
3T,
2T,
T, 0, T, 2T
.
R / 2 Hz) to
avoid creating ISI. Therefore to transmit a 1 MSPS signal over the
air, at least 1 MHz of RF frequency spectrum will be required. The
baseband filters will need a cutoff frequency of at least 500 kHz.
Notice that the frequency spectrum or bandwidth required
depends on the symbol rate,
R/2to
þ
the bit rate. We can have a much
higher bit rate, depending on the constellation type used. For
not
