DISCRETE-TIME CHANNELIZERS FOR AERONAUTICAL TELEMETRY: PART II VARIABLE BANDWIDTH

Transcription

1 DISCRETE-TIME CHANNELIZERS FOR AERONAUTICAL TELEMETRY: PART II VARIABLE BANDWIDTH Brian Swenson, Michael Rice Brigham Young University Provo, Utah, USA ABSTRACT A discrete-time channelizer capable of variable bandwidth operation and suitable for use in aeronautical telemetry is described. The channelizer is based on the polyphase filterbank implementation and is capable of selecting bandwidths that are odd multiples of MHz. The discrete-time approach reduces the number of continuous-time components and the need the maintain and calibrate these components. INTRODUCTION Channelization is the primary function of a telemetry receiver. A channelizer is most often thought of as a tunable device that selects a channel (defined by its center frequency and bandwidth) and passes only the energy in that channel to a demodulator. Note that to be of practical use, a telemetry channelizer must be capable of tuning to all the center frequencies in the assigned frequency band and be capable of selecting channels with different bandwidths. Most telemetry receivers currently in use employ a variation of the superheterodyne receiver to perform channelization. The basic premise of the superheterodyne receiver is to exploit the good properties of band-pass filters designed to operate at a fixed frequency by moving the signal to the filter using mixers to perform the required frequency translations. The superheterodyne receiver performs channel selection using coarsely tunable bandpass filters, frequency agile local oscillators, and banks of SAW filters centered at the last IF (typically 70 MHz). These receivers contain many continuous-time components that must be maintained and calibrated. A common method for reducing the cost and effort of maintaining the continuous-time components is to replace as many of them with discrete-time processors as possible. This paper develops a discrete-time version of the channelizer based on the polyphase filter bank. The fixed-bandwidth version of the channelizer was described in the companion paper []. Here the variable-bandwidth version, capable of selecting bandwidths that are odd multiples of MHz, is described.

2 Caveat lector this paper is a continuation of the companion paper [] and as such, cannot be read as a stand-alone paper for those new to the field of multirate discrete-time signal processing. AN EXAMPLE OF A DISCRETE-TIME CHANNELIZER WITH VARIABLE BANDWIDTH We introduce the concept of variable bandwidth multirate discrete-time channelizer using an example. Suppose there are three L-band ( MHz) signals on the air: an Mbit/s SOQPSK signal at MHz, a 500 kbit/s PCM/FM signal at MHz, and a 0 Mbit/s SOQPSK signal at 54.5 MHz. A representation of their spectra is illustrated in Figure (see point ). A low-noise block downconverter (LNB) is used to translate L-band to a lower frequency ( MHz) in preparation for sampling. The downconverted spectrum at the LNB output is illustrated at point in the figure. The LNB output is sampled at 360 Msamples/s to produce the discretetime spectrum illustrated at point 3 in the figure. As explained in the companion paper [], the discrete-time spectrum is partitioned into 360 segments of width /360 cycles/sample where each segment corresponds to a MHz channel in the continuous-time world. After sampling, the center frequencies the channels are indexed by the integer 0 k < 360 and are given by F 0 = k + / 360 cycles/sample, Ω 0 = πf 0 rads/sample. () As described in the companion paper [], an equivalent form of the channelizer is the bandpass filter, the heterodyne, and the downsampler shown inside the dashed box on the right-hand side of the block diagram at the top of Figure. The bandpass filter has a passband bandwidth equivalent to MHz. To select the PCM/FM signal at MHz, the channel index is set to k = 0. This centers the passband of the bandpass filter at the discrete-time frequency 0.5 /360 cycles/sample relative to the input sample rate as shown at point 4 in Figure. The bandpass filter attenuates all but the components corresponding to the PCM/FM signal at 0.5 /360 cycles/sample (and the accompanying in-band noise). (For more introductory information on how this channelizer works, see the discussion surrounding Figures 4 and 5 in the companion paper [].) The problem with this approach is when the bandwidth of the signal of interest is greater than the bandwidth of. This situation is illustrated by the bottom plot in Figure where the desired signal is the -Mbit/s SOQPSK signal. The bandpass filter is centered at the discrete-time frequency corresponding to MHz (the center frequency of the -Mbit/s SOQPSK signal) by setting the tuning parameter to k = 90. This places a copy of the too-narrow bandpass filter at the discrete-time frequency 90.5 /360 cycles/sample as shown. But this is not enough. If the filter is properly designed (more on this below), it is possible to select the entire signal by creating a bandpass filter whose transfer function is the combination of the seven bandpass filters as shown. The first key concept we exploit to design the variable-bandwidth discrete-time channelizer is stated as follows: Key Concept : Because filtering is a linear operation, the signal produced by a filter that is the sum of the seven filters is equal to the signal produced by the sum the outputs of the seven individual filters.

3 r(nt ) ➊ LNB ➋ ADC ➌ G(z) ❹ y(nt ) D y(mdt ) discrete-time channelizer e jω0n ➊ frequency (MHz) L-Band ➋ frequency (MHz) LNB Output ➌ frequency (cycles/sample) ,k = 0 ❹ (k = 84) (k = 86) (k = 88) (k = 90) (k = 9) (k = 94) (k = 96) Figure : A version of the discrete-time channelizer from the companion paper []. A block diagram of the channelizer is shown at the top. The remaining diagrams are representations of the spectra at different points in the system. 3

4 H 0(z M/D ) r(nt ) generalized commutator H (z M/D ). e jπ( k M + M ) y k (mdt ) e jπ( k M/D + M/D )m H M (z M/D ) phase shifting network e jπ(m )( k M + M ) Figure : A polyphase filterbank realization of the channelizer in the dashed box in Figure. This means if there were some efficient way to replicate an arbitrary number of the bandpass filters (7 in this example), then the variable bandwidth channelizer design would be straight forward. A remarkably efficient method for doing precisely this is identified in the second key concept. The second key concept involves the properties of the polyphase filterbank version of the discretetime channelizer. The polyphase filterbank version of the discrete-time channelizer was developed in [] and is described by the block diagram of Figure. This system replaces the blocks in the dashed box of Figure. In Figure, the integer M is the ratio of sample rate to channel bandwidth (that is /( ft ) = M = 360 and that is why the first Nyquist zone at the input sample rate is partitioned into 360 segments.) The downsample factor D determines the output sample rate (it is 360/D in this example) and partly determines the properties of the subfilters in the filterbank. To generate the polyphase filterbank of Figure, the downsample factor D must divide the integer M. The filter H(z) is the low-pass prototype filter whose frequency-translated version is the bandpass filter shown in the examples of Figure. The filter H (z) is obtained from H(z) by changing the sign on every other filter coefficient. The bank of multipliers applied to the subfilter outputs are phase shifters whose phase shift is defined by M, D and k, the channel selection parameter. This bank of multipliers is identified by the dashed box labeled phase shifting network in Figure For a each value of 0 k < M = 360, the phase shifters cophase the components in the subfilter outputs corresponding to the desired channel and cancel out the components corresponding to the other channels. When D < M not all of the channels wind up at baseband. The final multiplier performs a residual heterodyne operation that translates the signal components to baseband. The second key concept is derived from the observation that as signal processing moves from left to right, the first time the channel selection parameter k shows up is in the phase shifters. Consequently, the processing defined by the generalized commutator and polyphase filterbank must preserve all of the data from all M = 360 channels. Thus, we have the second key concept: 4

5 Key Concept : The signal components from all M channels is simultaneously available at the outputs of the M subfilters of the polyphase filterbank of Figure. This concept was illustrated for the case D = 90 in []. This illustration is repeated here in Figure 3 (a). Stated in more general language, the generalized commutator aliases all of the channels to one of M/D segments in the discrete-time frequency domain. The bandwidth of each segment is D F = D/M cycles/sample relative to the output sample rate. As a second example, the case for D = 5 is illustrated in Figure 3 (b). Here, all of the channels alias to one of M/D = 4 segments of bandwidth D F = D/M = /4 cycles/sample relative to the output sample rate. The channels whose index k reduces to 0 modulo 4 alias to the segment centered at 0.5 /4 cycles/sample relative to the output sample rate; the channels whose index k reduces to modulo 4 alias to the segment centered at.5 /4 cycles/sample relative to the output sample rate; and so on. The imagery of Figure 3 (b) suggests all of the signals possess the same bandwidth D F cycles/sample relative to the output sample rate. The more realistic scenario, represented by the running example, is portrayed in Figure 3 (c). The SOQPSK signal centered at MHz aliases to the channel indexed by k = 90 (that is, the signal spectrum is centered at 90.5 /360 cycles/sample relative to the input sample rate see Figure ). Because k mod (M/D) = 90 mod 4 = 8, this signal aliases to 8.5 /4 cycles/sample relative to the output sample rate and has a bandwidth that spans 3 channel segments as shown. Similarly, the PCM/FM signal centered at corresponds to the channel indexed by k = 0 (again, see Figure ) and because 0 mod 4 = 5, the signal aliases to 5.5 /4 cycles/sample relative to the output sample rate and has a bandwidth that spans channel segment. The SOQPSK signal centered at 54.5 MHz corresponds to the channel indexed by k = 9 and because 9 mod 4 = 3 the signal aliases to 3.5 /4 cycles/sample relative to the output sample rate and has a bandwidth that spans just under 3 channel segments. The main point is that copies of the bandpass filter may be produced in the polyphase filterbank realization by only replicating the phase shifting networks. To create a bank of bandpass filters centered at different frequencies, each phase shifting network uses a different value of k and all of the phase shifting networks process the same data. The overlap of the spectra in Figure 3 need not concern us because the phase shifting networks eliminate the undesired signal components. Combining Key Concept, Key Concept, and the bottom plot in Figure, we observe that a signal spanning 3 channel segments may be produced by applying 7 copies of the phase shifting network, each with a different value of k, to the subfilter outputs. This approach is illustrated in Figure 4. Here, the bank of phase shifters produce components of the SOQPSK signal centered at 90.5/360 cycles/sample relative to the input sample rate (see the bottom plot of Figure ). The end result is the series of samples representing the I/Q baseband version of the 3 MHz of bandwidth centered at MHz with a sample rate of 360/5 = 4 Msamples/s. PROTOTYPE FILTER DESIGN In its most basic form, the properties of the prototype lowpass filter H(e jω ) satisfy those in Figure 5 (a). Here, we have used the adjacent channel slot for the transition band. This is a reasonable 5

6 90 channels indexed by k mod 4 = alias to ⅝ cycles/sample 90 channels indexed by k mod 4 = 3 alias to ⅞ cycles/sample 90 channels indexed by k mod 4 = 0 alias to ⅛ cycles/sample 90 channels indexed by k mod 4 = alias to ⅜ cycles/sample (a) frequency (cycles/sample) frequency (cycles/sample) k = mod 4 k = 3 mod 4 k = 0 mod 4 k = mod 4 k = mod 4 (b) spectrum of the SOQPSK signal originally at spectrum of the SOQPSK signal originally at 54.5 spectrum of the PCM/FM signal originally at = = (c) Figure 3: A discrete-time frequency representation of the output of the filterbank shown in Figure : (a) the spectra for the case M = 360 and D = 90; (b) the spectra for the case M = 360 and D = 5; (c) the spectra for the case M = 360 and D 90 where the filterbank input consists of the three signal whose discrete-time spectra are shown at point 3 of Figure. 6

7 H 0(z 4 ). (k = 84) (k = 86) r(nt ) generalized commutator H (z 4 ). H 359(z 4 ) (k = 88) (k = 90) (k = 9) (k = 94) (k = 96) 8.5 jπ e 4 m y k (mdt ) Figure 4: Polyphase filterbank channelizer based on M = 360 and D = 5 and configured to select a bandwidth of 3/360 cycles/sample relative to the input sample rate centered at k = 90.5 /360 cycles/sample relative to the input sample rate. approach because it is rare that a signal is assigned to an adjacent channel in aeronautical telemetry applications. A narrower transition band may be used, but this requires a longer filter. When used in a variable bandwidth context, an additional property is required. The filter must have a transition band that exhibits the symmetry illustrated in Figure 5 (b). The symmetry conditions requires that the sum of the transition band and a reversed copy of itself sum to a constant. (The constant is here.) The formal definition of the symmetry is the following: = l= H ( e j(ω πl F )). () The individual summands for l =,, 0,, are plotted in Figure 5 (b). Note that the symmetry requirement means transition band of H(e jω ) must pass through 0.5 at Ω = π F. The filter response shown in Figure 8 of [] approximately meets these requirements. The filter was designed using the Parks-McClellan equiripple algorithm []. The filter design algorithm formed the basis for a design loop that iteratively adjusted the stop band to create a transition band whose magnitude was 0.5 at the desired frequency. (See Appendix A of [3] for a description of this approach to filter design applied to a slightly different application.) A filter designed using the Parks-McClellan algorithm is not guaranteed to have the desired symmetry in the transition band. But our experience with this approach is that the Parks-McClellan algorithm produces a very good approximation to the desired result. 7

8 pass band transition band stop band A 0 F 3 F frequency (cycles/sample) (a) j(ω+π4 F H e ) H j(ω+π F e ) H e jω j(ω π F H e ) j(ω π4 F H e ) 5 F 4 F 3 F F F 0 F F 3 F 4 F 5 F (b) frequency (cycles/sample) Figure 5: An illustration of the design parameters for the prototype low-pass filter: (a) the passband, transition-band, and stop-band definitions; (b) an illustration of the transition-band symmetry required to produce a composite band-pass filter. GENERALIZATIONS The general form of the discrete-time channelizer based on a polyphase filterbank is illustrated in Figure 6. The only difference between this channelizer and the fixed-bandwidth channelizer of Figure is how the subfilter outputs are processed. The fixed-bandwidth channelizer applies a single phase shifting network to the subfilter outputs whereas the variable-bandwidth channelizer applies multiple phase shifting networks to the subfilter outputs. The center frequency is defined by the parameter k and the bandwidth is defined by the number of filter banks and their associated indexes. Because the prototype filter has a bandwidth of F = ft cycles/sample at the input sample rate (this corresponds to f Hz), the discrete-time channelizer of Figure 6 is capable of selecting bandwidths that are odd multiples of f: in continuous time and B = (N ) f Hz N =,,... (3) BT = (N ) F cycles/sample N =,,... (4) in discrete time. To select a channel with bandwidth (N ) F cycles/sample centered on the channel indexed by k, the outputs of N phase shifting networks parameterized by the elements of the set K are summed. The N elements of the set K are { } K = k (N ), k (N ) +,..., k + (N ), k + (N ) (5) 8

9 H 0(z M/D ) r(nt ) generalized commutator H (z M/D ). N s y k (mdt ) e jπ( k M/D + M/D )m H M (z M/D ) M inputs N outputs Figure 6: The polyphase filterbank realization of the discrete-time channelizer capable of selecting channels with variable bandwidth. This is a generalization of the block diagram of Figure 4. where the special cases for N =, N =, and N = 3 are { } K = k, N = { } K = k, k +, N = { } K = k, k, k +, N = 3. (6) This shows that the phase shifter arrangement depends on the parity of N. When N is odd, the set K includes the index k and this applies the bandpass filter centered at (k + 0.5)/M as illustrated by the example in Figure 7 (a). When N is even, the set K skips the index k and applies the bandpass filters in the arrangement illustrated by the example in Figure 7 (b). Because the downsample factor D must divide the parameter M [], D cannot be selected arbitrarily. For the running of example using a sample rate of 360 Msamples/s and f = MHz, we have M = 360. The possible values for D are the divisors of 360. These values, along with corresponding output sample rate and usable RF bandwidth are listed in Table. We are now in a position to describe how the variable-bandwidth discrete-time channelizer configures itself. The procedure described below is based on the following assumptions: We assume the LNB output frequency is fixed and known to the internal configuration control. We assume the ADC sample rate /T and f are fixed in advance. This defines the parameters F and M which, in turn, set the requirements for the prototype low-pass filter H(z). This means H(z) is designed once by the receiver designer and is stored in the receiver. Thus The parity of an integer is its property of being even or odd. 9

10 as the telemetry engineer changes the center frequency and bandwidth settings, the prototype low-pass filter does not have to be recomputed. With these assumptions in place, the receiver configuration proceeds as follows:. User input: The telemetry engineer selects the center frequency f 0 (in Hz) and the bandwidth B (in Hz).. Calculate channel selection parameter k: After sampling the LNB output, express the desired center frequency as F 0 = F (k + /). From this, the parameter k is computed using k = F 0 F /. 3. Calculate down sample factor D: The downsample factor may be selected in a number of ways. An obvious approach is to find the largest divisor of M for which M/D is greater than the normalized bandwidth BT. See Table for an example for the case M = Set up phase shifting network: Express the normalized bandwidth BT (with units cycles/sample) as BT = (N ) F and calculate N using N = ( ) BT F +. The parameter N indicates how many phase shifting networks are needed whereas the parameters k and N determine which phase shifting networks are needed. 5. Configure the channelizer: Each new value of D requires a reconfiguration of the commutator, the spacing of the zeros in the subfilters of the polyphase filterbank, and the phase shifting network. In an FPGA implementation, this constitutes a new bit stream load that is easily accomplished in a few hundred milliseconds. If only the center frequency is changed (D remains fixed), then the only changes are a single parameter in the phase shifting network and the residual heterodyne. In a routine FPGA implementation, this can be accomplished by changing a register value and does not require a new bit stream load. 0

11 bandwidth = 9 F k 4 k k k + k +4 (k + 6) F (k + 5) F (k 4) F (k 5) F (k + 6) F (k + 5) F (k + 4) F (k + 3) F (k + ) F (k + ) F k F (k ) F (k ) F (k 3) F (k 4) F (k 5) F center frequency = (k + /) F center frequency = (k + /) F (a) bandwidth = 7 F k + k +3 k 3 k (k + 5) F (k + 4) F (k 3) F (k 4) F (k + 6) F (k + 5) F (k + 4) F (k + 3) F (k + ) F (k + ) F k F (k ) F (k ) F (k 3) F (k 4) F (k 5) F center frequency = (k + /) F center frequency = (k + /) F (b) Figure 7: Examples of the bandpass filter arrangements to select bandwidths of (N ) F for N odd and even: (a) N = 5, an example for odd N; (b) N = 4, an example for even N.

12 Table : Possible values for the downsample factor D for the discrete-time channelizer based on the polyphase filterbank shown in Figure 6 for an input sample rate of 360 Msamples/s and f = MHz. These values defined M = 360. maximum output passband D sample rate bandwidth (Msamples/s) (MHz)

13 COMMENTS ON IMPLEMENTING THE PHASE SHIFTING NETWORK The IFFT stuff goes here. CONCLUSIONS A variable-bandwidth version of the discrete-time channelizer developed in [] as been described. The channelizer is based on an RF architecture that uses an LNB to translate the block of channels in the desired frequency band (L-band, S-band, C-band, etc.) to a common frequency, and a discrete-time architecture that uses an ADC coupled with a polyphase filterbank whose parameters are easily adjusted to select different bandwidths centered at all the center frequencies available in the LNB output. The system is capable of selecting bandwidths that are odd multiples of MHz. REFERENCES [] B. Swenson and M. Rice. Discrete-time channelizers for aeronautical telemetry: Part I fixed bandwidth. In Proceedings of the International Telemetering Conference, Las Vegas, NV, October 0. [] A. Oppenheim and R. Schafer. Discrete-Time Signal Processing. Prentice-Hall, Upper Saddle River, NJ, 009. [3] M. Rice. Digital Communications: A Discrete-Time Approach. Pearson Prentice-Hall, Upper Saddle River, NJ,