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Chapter 11

11. PULSE SHAPING AND INTERPOLATION FILTERS

Different methods of designing the pulse shaping filters are reviewed in this chapter. In the digital modulators, phase distortion cannot be tolerated, thus the filters are required to have a linear phase response. A FIR filter can be guaranteed to have an exact linear phase response if the coefficients are either symmetric or antisymmetric about the center point. Three FIR filter structures (direct form, transposed direct form and hybrid form) are presented. The quantization effects and scaling methods within the fixed point FIR architectures are reviewed. Using canonic signed digit (CSD) coefficients, the FIR filtering operation can be simplified to add and shift operations. The well known carry save (CS) numbers are very attractive for VLSI implementation since the basic building block for arithmetic operations is a simple full adder. The multirate signal processing is particularly important in the digital modulator, where sample rates are low initially and must be increased for efficient subsequent processing. The efficient filter structures for the multirate signal processing (polyphase filters, halfband filters and comb filters) are presented. Taking advantage of the fact that in the modulator data streams in the I and Q paths are processed with the same functional blocks (see Figure 16-6), a further hardware reduction can be achieved by pipeline interleaving techniques.

11.1 Pulse Shaping Filter Design Algorithms

The pulse shaping filter design has two main objectives: minimization of the inter symbol interference (ISI) and maximization of the adjacent channel leakage power ratio (ACLR) [Che82], [Sam88], [Sam91], [Mor95]. [Sam88] and [Sam91] present two iterative algorithms that allow to the design of an overall filter of a given order N (an even number) with zero ISI and linear phase. The attenuation in the stopband is minimized. In [Sam88], only equiripple filters are considered and linear programming is used for deter-

220 Chapter 11

mining the filter taps. [Sam91] introduces a ripple weighting vector that allows arbitrary magnitude transfer functions to be designed, and uses equations instead of inequalities. Both [Sam88] and [Sam91] illustrate how to design a Nyquist filter that can be subsequently split into a transmitter and a receiver filter. [Sam88] and [Sam91] provide zero-ISI solutions but nonlinear phase characteristics. In [Mor95], the matched filter condition, i.e. identical transmit and receive filters with time reversal, is relaxed in order to obtain two linear phase transmit and receive filters and zero ISI in the composite filter. The transmit and the receive filters have different lengths. In [Che82], Chevillat and Ungerboeck present nonlinear optimization techniques for designing transmit and receive filters that result linear phase solutions with a non-zero-ISI [Che82].

The N-tap transmit filter is characterized by the coefficient vector h = (h0, h1, ..., hN-1)T, which is clocked at the rate M/ T corresponding to an oversampling ratio M. The receive filter (hr) is a K tap filter, which is M times over-sampled from the root raised cosine function. The transmit filter is convolved with the receive filter. Ideally, the result of the convolution will be an ideal raised cosine filter. There will be an EVM due to the truncation of the receive filter impulse response, if the length of the receive filter is short. It is therefore better to use a long receive filter so that the transmit filter will dominate the EVM.

The transmit and receive filter lengths are assumed to be either even or odd, so as to have one middle sample for decision in the composite pulse RC(n). The convolution of the transmitter and receiver filters should satisfy

where nc is the center tap and M is the over-sampling ratio. The center tap is (N+K-2)/2. The total number of the terms in (11.1) is 2L, where L = ¬nc/M¼ and ¬x¼ denotes the integer part of x. The equation (11.1) can be written as

where the elements of the "shift" matrices Sl are zero, except si k(l) = 1 for i k = (N K)/2 + Ml [Che82]. The "shift" matrices Sl are N K matrices.

The passband ripples of the linear phase half-band filters (interpolation filters in Figure 16-6) cause EVM as well, which could be partly compensated for by predistortion of the pulse shaping filter. The receive filter (hr) could be convolved with the interpolation filters. This convolution could be calculated with the noble identities [Vai93]. The result is decimated back to the M over-sampled ratio and convolved with the transmit filter in (11.2).

One code channel is transmitted, when the EVM is measured. The EVM consists of two components, which are mutually uncorrelated:

l L l≠0

whereδ e2 is the quantization noise due to finite word length effects. The D/A converter dominates this quantization noise, because it is the most critical component. The ISI term is

and W is a N N matrix. A linear constraint is added to guarantee proper scaling of the pulse peak

The lowpass channel energy (Ec) from dc to f (the cut-off frequency of the lowpass channel) is