Understanding Square Root Raised Cosine Spectrum (3).pdf

UNDERSTANDING SQUARE ROOT RAISED COSINE SPECTRUM

1. Understanding Square Root
Raised Cosine Spectrum
PRESENTED BY
UDAY KIRAN REDDY
LAKSHMANA RAO
MAHESH
ABHISHEK
A Pulse Shaping Technique

2. Introduction
Pulse shaping is a fundamental concept in digital communication systems that involves
modifying the shape of transmitted pulses to achieve specific objectives. The primary
goals of pulse shaping are to control the spectral characteristics of the transmitted
signal and minimize inter-symbol interference (ISI). Here's a brief overview of pulse
shaping in digital communication:

3. Square-Root Raised Cosine
Signals (SRRC)
One of the main drawbacks of all the signal waveforms is that although they
can very well control the power emissions within the bandwidth of interest,
they send relatively high amounts of power out of this one. A practical way of
reducing the side-lobes of the spectrum of the navigation signals could be to
use a Raised Cosine Filter (RCF) since this has a limited bandwidth. The
Raised Cosine Filter is a particular case of Nyquist filter and is defined in the
frequency domain as follows:

4. where W-W0 is defined as the excess bandwidth and indicates how much the spectrum
of the Raised Cosine spills over a given bandwidth W0. As we know, Nyquist pulses
(filters) are pulses that result in no Inter Symbol Interference (ISI) at the sampling time.
The Nyquist pulse-shaping criterion or Nyquist condition for zero ISI is fulfilled if
where
Indeed, this is a necessary and sufficient condition which can also be expressed as follows

5. where X(f) is the Fourier transform of a generic signal x(t) and Tc the time period of the pulse. The
Raised Cosine filter that we described some lines above has an equivalent representation in the
time domain. This is shown to be:
Another way of expressing the excess bandwidth is by means of the roll-off factor, which is
defined as follows:

6. fig 1

7. FIG 2

8. It is important to note that a band-limited Nyquist pulse cannot avoid by itself the ISI unless
the channel is ideal. This means that the RC pulses have got to be implemented together
with an equalizer at the receiver for the correct identification of the symbols at the sampling
time. We can express this in the following expression:
where Hrc(f) is the transmission filter, Hc(f) is the channel frequency response, Hrx(f) is
the receiver filter and He(f) is the equalizer. The usual approach is to design the
transmitter and receiver filters such that

9. and leave the equalizer filter to take care of the imperfections and ISI caused by the channel:
Moreover, it can be shown that
where we can recognize that the bilateral bandwidth is finite and of value ((1+alpha)/Tc). In the same
manner, the time representation of such SRRC pulses is shown to adopt the following form .

10. Applications:
1.SRRC pulse shaping is widely used in digital communication systems, especially in
scenarios where bandwidth efficiency and reduced ISI are critical.
2.Commonly employed in systems utilizing techniques like Quadrature Amplitude
Modulation (QAM) and Phase Shift Keying (PSK)
3.One of the most important disadvantages is the fact that the RC signal is handicapped
from the beginning regarding its potential improvement of performance.
4.Another consequence of the fact that the SRRC modulation is bandlimited is that its
auto-correlation function has a very rounded peak.

11. THANK YOU !