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Digital Communications. EE549/449 FALL 2001 Lecture #26 Pulse Shaping Controlled Intersymbol Interference Wednesday October 24, 2001. Root Raised Cosine (RC) rolloff Pulse Shaping.

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Digital communications l.jpg
Digital Communications

EE549/449 FALL 2001

Lecture #26

Pulse Shaping

Controlled Intersymbol Interference

Wednesday October 24, 2001

Root raised cosine rc rolloff pulse shaping l.jpg
Root Raised Cosine (RC) rolloff Pulse Shaping

  • 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

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Practical Issues with Pulse Shaping

  • 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

  • RC rolloff pulses are less sensitive to timing errors than Sa(.) pulses

    • Larger values ofare 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

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  • 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

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Controlled ISI

  • 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

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Partial Response Signaling (PRS)

  • 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

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



  • 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

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Duobinary Signaling

  • 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

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  • 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:

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  • 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:

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  • Duobinary Decoding:

    • 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)

  • Slide13 l.jpg

    • In general, ( summation followed by rectangular low pass filteringM-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

    Slide14 l.jpg

    A Duo-binary Baseband System summation followed by rectangular low pass filtering

    • 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)

    Slide15 l.jpg

    • Require more power summation followed by rectangular low pass filtering

      • 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


      • Problem 3 can be solved by a technique known as precoding

      • Problem 4 is solved by a technique known as modified duobinary

    Summary of duobinary baseband system l.jpg
    Summary of Duobinary Baseband System summation followed by rectangular low pass filtering

    • 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

    Slide18 l.jpg

    Composite pulses arising from like and unlike summation followed by rectangular low pass filtering

    combinations of input impulse pair

    Slide19 l.jpg

    Duobinary waveform arising from an example binary sequence summation followed by rectangular low pass filtering

    Example 30 duobinary coding l.jpg
    Example 30: (Duobinary Coding) summation followed by rectangular low pass filtering

    (See example 2.4)

    • Binary sequences xk 0 0 1 0 1 1 0

    • Amplitude: ak 1 -1 -1 1 -1 1 1 -1

    • Coding Rule:

    • Decoding Rule:

    • Output sequence