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Figure 5.9 Desired amplitude response. in Software Embed EAN-13 Supplement 2 in Software Figure 5.9 Desired amplitude response.

Figure 5.9 Desired amplitude response. using barcode integrating for software control to generate, create ean13 image in software applications. QR Code i i i i i Tranter Book 2003/11/18 16:12 page 166 #184. 166 1, 0,. Filter Models and Simulation Techniques 5 . Ad (f ) =. f . < 0.1, 0.25 < f < 0.35 0.12 < f < 0.23, 0.37 < f < 1 (5.58). The three fr equency bands not included in the expression for Ad (f ) are transition bands. These are indicated by the heavy black regions in Figure 5.9.

The MATLAB program for designing the lter is as follows: % File: c5 yw.m order = 20, f = [0 0.1 0.

12 0.23 0.25 0.

35 0.37 1]; amp = [1 1 0 0 1 1 0 0]; [b,a] = yulewalk(order,f,amp); freqz(b,a) % End of script file. % % % % % degree of polynomials frequency points amplitude response synthesize filter display results.

Note that a Software GS1-13 20th order lter is generated so that both the denominator and the numerator polynomials of H(z) are polynomials of degree 20 in z 1 . Executing the program yields the result illustrated in Figure 5.10.

Note that the stopband attenuation is approximately 30 dB or better. The passbands are reasonably well shaped and the phase response is approximately linear across the passbands. Whether or.

20 0 -20 -40 -60 -80 0 0.1 0.2 0.

3 0.4 0.5 0.

6 0.7 Normalized frequency (Nyquist == 1) 0.8 0.

9 1. M agnitude R Software EAN13 esponse (dB ) P hase (degrees). -1500 0 0.1 0.2 0.

3 0.4 0.5 0.

6 0.7 Normalized frequency (Nyquist == 1) 0.8 0.

9 1. Figure 5.10 Computer-aided design results for IIR lter. i i i i i Tranter Software EAN-13 Supplement 5 Book 2003/11/18 16:12 page 167 #185. Section 5.5. FIR Filters: Synthesis Techniques and Filter Characteristics not this lt er is a satisfactory approximation to the ideal response de ned by (5.58) is a matter of judgment, and a system simulation using this lter may be required to answer that question. Increasing the lter order will improve the approximation error but at the cost of increased complexity resulting in increased simulation runtime.

. Error Sources in IIR Filters In this sect ion we summarize the error sources resulting in the approximation of analog lters by IIR digital lters. We have seen that the source of error is dependent upon the synthesis method used. For the most part these errors may be minimized by using a very high sampling frequency in the simulation.

However, unnecessarily high sampling frequencies result in simulations that take an unnecessarily long time to execute. As is usually the case, selection of a sampling frequency for a simulation involves a tradeo between accuracy and the time required to execute the simulation. The error sources are summarized in Table 5.

1.. Table 5.1 IIR Digital Filter Error Sources Synthesis Technique Impulse invariant Step invariant Bilinear z-transform Error Source Aliasing Aliasing Frequency warping CAD method Approximation error To Mimimize Error Choose a higher sampling frequency Choose a higher sampling frequency Select a sampling frequency much larger than the highest critical frequency Increase lter order or use another synthesis method more suitable to the application. FIR Filters: Synthesis Techniques and Filter Characteristics If the impul se response of a lter h[n] is nite, or has been made nite by truncating an originally in nite duration impulse response, the lter output in the discrete time domain is given by. y(nT ) = T h(kT )x{(n k)T }. (5.59). i i i i i Tranter EAN-13 for None Book 2003/11/18 16:12 page 168 #186. Filter Models and Simulation Techniques 5 . which, using standard DSP notation, is y[n] =. h[n]x[n k] =. bk x[n k]. (5.60). or, in terms EAN-13 for None of the z transform,. Y (z) =. bk z k X(z). (5.61). Equation (5. European Article Number 13 for None 60), which de nes an FIR digital lter, is the discrete-time version of the convolution integral for analog (continuous time) lters. Note that (5.

60) and (5.61) are the same as the lter model de ned by (5.1) and (5.

3) with ak = 0 for k 1. The lter is FIR (Finite duration Impulse Response) since there are at most N + 1 nonzero terms in h[n]. Since ak = 0 for k 1 the FIR lter has no feedback paths.

The convolution operation in time domain de ned in (5.60) can be simulated using the MATLAB function filter. The FIR lter is attractive for a number of reasons, the most signi cant of which are as follows: 1.

Not all lters in a communication system can be expressed in terms of a transfer function in the Laplace transform domain and hence the IIR techniques described in the previous sections cannot be applied directly. Two important lters that fall into this category are the square root raised cosine (SQRC) pulse shaping lter, and the Jakes Doppler lter. These lters are easily simulated using the FIR approach.

2. In many simulation applications, lter data may be given empirically in the form of measured frequency response or impulse response data. It is much easier to simulate these lters using the FIR approach.

While there are some techniques available for tting an ARMA model to the frequency response data, such approximations do not yield satisfactory results when the frequency response data is not relatively smooth. 3. With the FIR approach we can specify arbitrary amplitude and phase responses, and they can be independent of each other.

Thus it is possible, for example, to simulate an ideal brick-wall lter with linear phase response. 4. The FIR lters lack feedback and are therefore always stable The FIR simulation model does have one main drawback, namely, it is computationally not as e cient as the IIR implementation.

Direct implementation of the convolution operation given in (5.60) requires N (complex) multiplication and additions for each output sample to be generated. If the impulse response is very long, say N > 1024 points, the FIR algorithm will execute much more slowly than a typical IIR lter algorithm.

For an IIR lter the number of additions and multiplications required to generate each output sample are determined by the order.
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