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Digital Communications

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EE549/449 FALL 2001

Lecture #26

Pulse Shaping

Controlled Intersymbol Interference

Wednesday October 24, 2001

- We will see later in the semester that the noise is minimized at the receiver by using a matched filter
- If the transmit filter is H(f), then the receive filter should be H*(f)

- The combination of transmit and receive filters must satisfy Nyquist’s first method for zero ISI
- Transmit filter with the above response is called the root raised cosine rolloff filter
- Root Raised Cosine rolloff pulse shapes are used in many applications such as US Digital Cellular, IS-54 and IS-136

- Like the Sa(.) pulse, RC rolloff pulses extend infinitely in time
- However, a very good approximation can be obtained by truncating the pulse
- E.g., we can make h(t) extend from -3Tb to +3Tb

- However, a very good approximation can be obtained by truncating the pulse
- RC rolloff pulses are less sensitive to timing errors than Sa(.) pulses
- Larger values ofare more robust against timing errors

- US Digital Cellular (IS-54 & IS-136) uses root RC rolloff pulse shaping with = 0.35
- IS-95 uses pulse shape that is slightly different from RC rolloff shape
- European GSM uses Gaussian shaped pulses

- Implementation of Raised Cosine Pulse:
- Practical pulses must be truncated in time
- Truncation leads to sidelobes - even in RC pulses

- Can be digitally implemented with an FIR filter
- Analog filters such as Butterworth filters may approximate the tight shape of this spectrum
- Sometimes a “square-root” raised cosine spectrum is used when identical filters are implemented at transmitter and receiver
- This has to do with matched filtering

- Practical pulses must be truncated in time

- To achieve zero ISI, we have seen that it is necessary to transmit at below the Nyquist rate
- Is it possible to relax the condition on zero ISI and allow for some amount of ISI in order to achieve a rate > 2B?
- Idea is to introduce some controlled amount of ISI instead of trying to eliminate it completely
- ISI that we introduce is deterministic (or controlled) and hence we can take care of it at the receiver
- How do we do this?
- Controlled amount of ISI is introduced by combining a number of successive binary pulses prior to transmission
- Since the combination is done in a known way, the receiver can be designed to correctly recover the signal

- We will now discuss different methods of controlled ISI

- Also known as Doubinary signaling, Correlative coding, Polybinary
- PRS is a technique that deliberately introduces some amounts of ISI into the transmitted signal in order to ease the burden on the pulse-shaping filters
- It removes the need to strive at achieving Nyquist filtering conditions, and high rolloff factors
- This strategy involves two key operations
- Correlative Filtering (CF)
- Digital Precoding (DP)

- CF purposely introduces some ISI, resulting in a pulse train with higher amplitude levels and correlated amplitude sequences
- Nyquist rate no longer applies since the correlated symbols are no longer independent

- Hence higher signaling rate can be used

- The transfer function H(f) is equivalent to the Tap Delay Line model

- Since h(t) = sinc(t/T) and R=1/T, the overall impulse response is
and

where

- PRS changes the amplitude sequence ak a+k
- a+k has a correlated amplitude span of N symbols since each a+k depends on the previous N values of ak
- Also, when ak has M levels, a+k sequence has M+ > M levels
- A whole family of PRS methods exists
- Lets look at a few specific cases of PRS

- Also called class 1 signaling
- Simplest form of PRS with M = 2, N = 1, Co = C1 = 1
- The input data sequence is combined with a 1-bit delayed version of the same sequence (the controlled ISI) and then passed through the pulse-shaping filter
- Duobinary Encoder

- Each incoming pulse is added to the previous pulse
- The bit or data sequence {yk} are not independent
- Each yk digit caries with it the memory of the prior digit

- It is this correlation between digit that is considered the controlled ISI which can be easily removed at the receiver
- Impulse Response of Duobinary Signal:

- From
it can be shown that (exercise - show this)

- Impulse response h(t) for the duobinary scheme is simply the sum of two sinc waveforms, delayed by one bit period w.r.t each other:

- Duobinary signaling can be interpreted as adjacent pulse summation followed by rectangular low pass filtering
- Encoder takes a 2 level waveform and produces a 3 level waveform

- The role of the receiver is to recover xk from yk
- Transmitted signal (assuming no noise) is
- xk can assume one of 2 values A, depending on whether the k-th bit is 1 or 0
- Since yk depends on xk and xk-1, yk can have 3 values (no noise)

- In general, (M-ary transmission), PRS results in 2M-1 output levels
- Detection involves subtracting xk-1 decisions from yk digits such that
- The detection process is the reverse operation at the transmitter
- Decision rules is
- A major drawback to this technique is that once errors are made, they tend to propagate through the system

A Duo-binary Baseband System

- Advantage:
- It permits transmission at the Nyquist rate without the need for linear phase rectangular pulse shaping

- Disadvantages:
- There is no one to one mapping between detected ternary symbol and the original binary digits (2 3)

- Require more power
- Ternary nature of duobinary signal requires about 3 dB greater SNR compared to ideal signaling (i.e, binary) for a given PB

- The decoding process results in propagation of errors
- Because output data bits are decoded using previous data bit, if it is in error then the new output will be in error, and so on
- In other words, errors will propagate through the system

- It is ineffective for AC coupled signal
- PSD has substantial values at zero making it unsuitable for use with AC coupled transmission
Note:

- Problem 3 can be solved by a technique known as precoding
- Problem 4 is solved by a technique known as modified duobinary

- PSD has substantial values at zero making it unsuitable for use with AC coupled transmission

- In general, (M-ary transmission), PRS results in 2M-1 output levels
- Detection involves subtracting xk-1 decisions from yk digits such that
- Decision rules is

Composite pulses arising from like and unlike

combinations of input impulse pair

Duobinary waveform arising from an example binary sequence

(See example 2.4)

- Binary sequencesxk 0 0 1 0 1 1 0
- Amplitude: ak1 -1 -1 1 -1 1 1 -1
- Coding Rule:
- Decoding Rule:
- Output sequence