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α
reflects the compromise between the signal
occupied bandwidth and the power parameter in the
frequency response shape of the filter
Signal occupied bandwidth = symbol rate X
(1+α)
symbols.
Filters with rectangular frequency responses are not possible in reality and filters with infinitely
steep responses require infinite number of poles. However, we can use the raised cosine filter (RCF)
to complete the zero-crossing Nyquist criterion. When these filters are used, the roll-off between the
passband and the stopband is a cosinc function. For such filters, “half” are placed in the transmitters
and the other “half” are put in the receivers. In this way, the entire raised cosine response can be
obtained from the Root RCF used simultaneously by the transmitter and the receiver. In addition to
noise filtering, the receiver can also use RCF to make decisions at the maximum signal-to-noise ratio,
which is the characteristic of matched filter pairs.
Nyquist filter coefficient α
Channels with ideal low-pass characteristics are difficult to be implemented and channels with
roll-off characteristics are practically used. The increase of its channel bandwidth over Nyquist
bandwidth is: Roll-off factor α. It can be expressed as α = (BW − f N)/ f N, where BW is the channel
bandwidth and f N is the Nyquist bandwidth. α is a value between 0 and 1. 1+α indicates the extra
bandwidth required for the filter to send signals without ISI. α is also called “excessive bandwidth
factor”, which means that the required occupied bandwidth exceeds the ideal occupied bandwidth at
the same symbol rate. The bandwidth occupied by RCF-filtered signals is approximately (1+α)Rs,
where Rs is the symbol rate. As shown in the Attached Figure 10, the occupied bandwidth BW is Rs
(because Fs = 0.5Rs, BW = 2Fs) when α=0; the occupied bandwidth is 2Rs when α=1. The typical
value of α is usually between 0.35 and 0.5. The above figure shows the frequency characteristics of
the RCF with different roll-off factors α. If the roll-off factor α is smaller, the filter characteristics are
steeper, but the sidelobe level generated by the pulse is higher, causing the higher requirements for
the sampling timing tolerance. These filters are generally implemented with finite impulse response
(FIR) filters in some DSPs. The raised cosine roll-off filters have been widely used due to their
abilities to transmit signals with great power and fast convergence to reduce inter-symbol
interference. It can be seen that the Nyquist filter requires the occupied bandwidth greater than the
symbol rate Rs and that its band utilization is no longer optimal. A Gaussian filter, not a Nyquist filter,
is used in the GSM system. The corresponding “excessive bandwidth factor” is BT (bandwidth time
factor), whose value is usually between 0.3 and 0.5. The modulation symbol rate of GSM is 270.833
KHz and the channel bandwidth is 200 KHz, so the channel bandwidth is smaller than the modulation
symbol rate.
Attached Figure 11 Diagram of Relation between Channel Bandwidth and α
