Square Root Raised Cosine Filter

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Square Root Raised Cosine Filter

Digital Communication, 4th Edition Chapter 9: Signal Design for Band-Limited Channels John G. Proakis

We consider the problem of signal design when the channel is band-limited to some specified bandwidth of W Hz. The channel may be modeled as a linear filter having an equivalent low-pass frequency response C( f ) that is zero for | f | >W. Our purpose is to design a signal pulse g(t) in a linearly modulated signal, represented as v ( t ) = ∑ I n g ( t − nT ) n

that efficiently utilizes the total available channel bandwidth W. When the channel is ideal for | f | ≤W, a signal pulse can be designed that allows us to transmit at symbol rates comparable to or exceeding the channel bandwidth W. When the channel is not ideal, signal transmission at a symbol rate equal to or exceeding W results in inter-symbol interference (ISI) among a number of adjacent symbols. 2

Characterization of Band-Limited Channels
For our purposes, a band-limited channel such as a telephone channel will be characterized as a linear filter having an equivalent low-pass frequency-response characteristic C( f ), and its equivalent low-pass impulse response c(t). Then, if a signal of the form

s (t ) = Re v(t )e j 2πf ct



is transmitted over a band-pass telephone channel, the equivalent low-pass received signal is

rl (t ) = ∫ v(τ )c(t − τ )dτ + z (t ) = v(t ) ∗ c(t ) + z (t ) 3

where z(t) denotes the additive noise.

Characterization of Band-Limited Channels
Alternatively, the signal term can be represented in the frequency domain as V( f )C( f ), where V( f ) = F[v(t)]. If the channel is band-limited to W Hz, then C( f ) = 0 for | f | > W. As a consequence, any frequency components in V( f ) above | f | = W will not be passed by the channel, so we limit the bandwidth of the transmitted signal to W Hz. Within the bandwidth of the channel, we may express C( f ) as C ( f ) = C ( f ) e jθ ( f ) where |C( f )|: amplitude response. ( f ): phase response. The envelope delay characteristic: τ ( f ) = − 1 dθ ( f ) 2π df 4

Characterization of Band-Limited Channels
A channel is said to be nondistorting or ideal if the amplitude response |C( f )| is constant for all | f | ≤ W and ( f ) is a linear function of frequency, i.e., ( f ) is a constant for all | f | ≤ W. If |C( f )| is not constant for all | f | ≤ W, we say that the channel distorts the transmitted signal V( f ) in amplitude. If ( f ) is not constant for all | f | ≤ W, we say that the channel distorts the signal V( f ) in delay. As a result of the amplitude and delay distortion caused by the nonideal channel frequency-response C( f ), a succession of pulses transmitted through the channel at rates comparable to the bandwidth W are smeared to the point that they are no longer distinguishable as well-defined pulses at the receiving terminal. Instead, they overlap, and thus, we have inter-symbol interference (ISI). 5

Characterization of Band-Limited Channels
Fig. (a) is a band-limited pulse having zeros periodically spaced in time at T, 2T, etc. If information is conveyed by the pulse amplitude, as in PAM, for example, then one can transmit a sequence of pulses, each of which has a peak at the periodic zeros of the other pulses.


Characterization of Band-Limited Channels
However, transmission of the pulse through a channel modeled as having a linear envelope delay ( f ) [quadratic phase ( f )] results in the received pulse shown in Fig. (b), where the zerocrossings that are no longer periodically spaced.


Characterization of Band-Limited Channels
A sequence of successive pulses would no longer be distinguishable. Thus, the channel delay distortion results in ISI. It is possible to compensate for the nonideal frequency-response of the channel by use of a filter or equalizer at the demodulator. Fig. (c) illustrates the output of a linear equalizer that compensates for the linear distortion in...
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