Christian Mastromattei, Jimmy Zhan
A report submitted to the Department of Electrical and Computer Engineering in conformity
with the requirements of ELEC 486
Faculty of Applied Science and Engineering
Kingston, Ontario, Canada
April 16, 2013 Introduction:
Bandwidth (BW) is one of the most crucial resources in the communication industry. Fiber optic
communication is band limited and typically occurs over the C-band. As modulation rates for
optical signals increase each channel occupies more BW in the optical band, meaning less
channels will be available at higher modulation rates. In order to squeeze more channels together
over a band, pulse shaping can be done. Pulse shaping is a process by which an optical signal is
shaped such that its BW is reduced in the frequency domain. Various techniques for pulse
shaping are used, with the objective of limiting the bandwidth of each channel while eliminating
Intersymbol Interference (ISI) and crosstalk.
The Rectangular Pulse:
The rectangular pulse is the most basic information unit in a digital transmission scheme. It has a
defined amplitude and period. A sequence of such pulses constitutes the transmission of
information. The information is encoded in the amplitude of the pulse. The simplest case is when
a binary 0 is encoded as the absence of a pulse (A = 0) and a binary 1 is encoded as the presence
of a pulse (A = constant). Since each pulse spans the period T, the maximum pulse rate is 1/T
Hz. The pulse amplitude can be more sophisticated and take on multiple discrete levels
(including negative values), so that each pulse can represent more than one bit  Figure. A rectangular pulse, with period T0, and amplitude A.
In these cases where multiple amplitudes and/or multiple simultaneous pulses transmit a single
unit of data, each unit of data (or symbol) represent multiple bits. The group of bits that a single
unit of data represents is defined as symbol. The unit symbols per second, is defined as the baud
rate. The data transmission bits rate is the baud rate multiplied by the number of bits represented
by each symbol . This means that a lower transmission rate can be used to transmit symbols as
opposed to directly transmitting bits, which is the primary reason that the more sophisticated data
transmission systems .
Spectrum of a Rectangular Pulse:
The Fourier Transform (spectrum) of the rectangular pulse is a sinc function, which is sin(x)/x.
This result is the same for any amplitude or frequency of the rectangular pulse. Although the
amplitude of the rectangular pulse does proportionally affect the magnitude of the frequency
Figure. The frequency spectrum of a rectangular pulse, note the null points at integer multiples of
The sinc function is characterized by these peaks extending from negative infinity to positive
infinity, but an important observation is that the null points (where the spectral magnitude is zero) always occurs at integer multiples of f0 (1/T), which is the symbol rate. These null points
are therefore solely determined by the pulse period. If two sinc functions which have the same
period were to add together, shifted by an integer multiple of the period, sampling only at the
null points would yield an undistorted signal. This result will be used to eliminate Intersymbol
Interference (ISI) during pulse shaping .
In general optical signals can be shaped in the electrical domain before modulation, or optically
after modulation. For QPSK systems, pulse shaping is done pre-modulation in digital signal
processing (DSP). There are two methods used to minimize the spacing between consecutive
channels; orthogonal frequency-division multiplexing (OFDM) and Nyquist wavelength division
multiplexing (N-WDM). The latter is more commonly implemented in optical systems since it is
easier to realize  and is the main focus of this report.
Nyquist Wavelength Division Multiplexing:
The ideology behind N-WDM is simple. All of the important data in a modulated optical signal
is contained in the main lobe of its spectra, which has width equal to the modulation frequency.
If each channel can be band-pass filtered such that the BW of the filter is the modulation
frequency, then any redundant frequencies of the signal spectra in the channel can be eliminated
and only the main lobe of the signal will remain. Theoretically this would allow all of the
channels to be as tightly squeezed together as possible since they would be separated by the
modulation rate; there would be no excess or wasted BW between consecutive channels (see
Figure 1). One would need an ideal sinc-filter to implement this, which is impossible to do in
practice. There are a few filters that can be used instead of the sinc filter; the raised-cosine filter
is commonly used. Figure 1: Ideal Nyquist wavelength division multiplexing in the frequency and time
domain. Note the infinite response in the time domain 
Raised Cosine Filter:
The raised cosine filter is the most commonly used filter for pulse shaping in communication
systems, especially multichannel communications that have channel modulation frequencies
close together. The main benefits of this filter are that it can eliminate ISI, since it satisfies
Nyquist ISI criterion (see Figure 2). It has a configurable excess bandwidth, which depends on
the factor β. The transfer function of the filter is given by:
where β is the roll-off factor and T is the symbol period. The roll-off factor corresponds to a
measure of the excess bandwidth of the filter. The Nyquist bandwidth of the filter is given by
1/2T. Figure 2: Illustration of the ISI Nyquist criteria. If sampling occurs and integer multiples
of T, then the signal from previous and successive signals are zero. 
Figures 3 and 4 show the raised-cosine filter in the frequency and time domain respectively. As β
approaches 0, the raised-cosine filter becomes a sinc filter. One can see that the larger β
becomes, the more BW the filtered signal will have, and thus it is advantages to try to decrease β.
Figure 3: Frequency spectra of a raised cosine filter for various roll-off factor values Figure 4: Impulse response of a raised cosine filter for various roll-off factor values
There are three important points in the frequency spectrum associated with raised cosine filters.
The first is the Nyquist frequency, which occurs at f0/2 (1/2T). This is the minimum possible
bandwidth that can be used to transmit data without loss of information. As shown in Fig.3 this
bandwidth corresponds to a perfect (β=0) filter. Note that the response crosses through the half
amplitude at this point regardless of β value. The second important point is the stop band
frequency (fstop), defined as the frequency at which the response first reaches zero magnitude.