DISCRETE-TIME CHANNELIZERS FOR AERONAUTICALTELEMETRY: PART II — VARIABLE BANDWIDTH

Brian Swenson, Michael RiceBrigham Young University

Provo, Utah, USA

ABSTRACT

A discrete-time channelizer capable of variable bandwidth operation and suitable for use in aero-nautical telemetry is described. The channelizer is based on the polyphase filterbank implemen-tation and is capable of selecting bandwidths that are odd multiples of 1 MHz. The discrete-timeapproach reduces the number of continuous-time components and the need the maintain and cali-brate these components.

INTRODUCTION

Channelization is the primary function of a telemetry receiver. A channelizer is most often thoughtof as a tunable device that selects a channel (defined by its center frequency and bandwidth) andpasses only the energy in that channel to a demodulator. Note that to be of practical use, a telemetrychannelizer must be capable of “tuning” to all the center frequencies in the assigned frequency bandand be capable of selecting channels with different bandwidths.

Most telemetry receivers currently in use employ a variation of the superheterodyne receiver toperform channelization. The basic premise of the superheterodyne receiver is to exploit the goodproperties of band-pass filters designed to operate at a fixed frequency by “moving the signalto the filter” using mixers to perform the required frequency translations. The superheterodynereceiver performs channel selection using coarsely tunable bandpass filters, frequency agile localoscillators, and banks of SAW filters centered at the last IF (typically 70 MHz). These receiverscontain many continuous-time components that must be maintained and calibrated.

A common method for reducing the cost and effort of maintaining the continuous-time componentsis to replace as many of them with discrete-time processors as possible. This paper develops adiscrete-time version of the channelizer based on the polyphase filter bank. The fixed-bandwidthversion of the channelizer was described in the companion paper [1]. Here the variable-bandwidthversion, capable of selecting bandwidths that are odd multiples of 1 MHz, is described.

1

Caveat lector — this paper is a “continuation” of the companion paper [1] and as such, cannot beread as a stand-alone paper for those new to the field of multirate discrete-time signal processing.

AN EXAMPLE OF A DISCRETE-TIME CHANNELIZERWITH VARIABLE BANDWIDTH

We introduce the concept of variable bandwidth multirate discrete-time channelizer using an exam-ple. Suppose there are three L-band (1435 – 1535 MHz) signals on the air: an 11 Mbit/s SOQPSKsignal at 1485.5 MHz, a 500 kbit/s PCM/FM signal at 1496.5 MHz, and a 10 Mbit/s SOQPSKsignal at 1514.5 MHz. A representation of their spectra is illustrated in Figure 1 (see point 1). Alow-noise block downconverter (LNB) is used to translate L-band to a lower frequency (400 – 500MHz) in preparation for sampling. The downconverted spectrum at the LNB output is illustratedat point 2 in the figure. The LNB output is sampled at 360 Msamples/s to produce the discrete-time spectrum illustrated at point 3 in the figure. As explained in the companion paper [1], thediscrete-time spectrum is partitioned into 360 segments of width 1/360 cycles/sample where eachsegment corresponds to a 1 MHz channel in the continuous-time world. After sampling, the centerfrequencies the channels are indexed by the integer 0 ≤ k < 360 and are given by

F0 =k + 1/2

360cycles/sample, Ω0 = 2πF0 rads/sample. (1)

As described in the companion paper [1], an equivalent form of the channelizer is the bandpass fil-ter, the heterodyne, and the downsampler shown inside the dashed box on the right-hand side of theblock diagram at the top of Figure 1. The bandpass filter has a passband bandwidth equivalent to 1MHz. To select the PCM/FM signal at 1496.5 MHz, the channel index is set to k = 101. This cen-ters the passband of the bandpass filter G(ejΩ) at the discrete-time frequency 101.5/360 cycles/samplerelative to the input sample rate as shown at point 4 in Figure 1. The bandpass filter attenuates allbut the components corresponding to the PCM/FM signal at 101.5/360 cycles/sample (and the accom-panying in-band noise). (For more introductory information on how this channelizer works, seethe discussion surrounding Figures 4 and 5 in the companion paper [1].)

The problem with this approach is when the bandwidth of the signal of interest is greater thanthe bandwidth of G(ejΩ). This situation is illustrated by the bottom plot in Figure 1 where thedesired signal is the 11-Mbit/s SOQPSK signal. The bandpass filter is centered at the discrete-timefrequency corresponding to 1485.5 MHz (the center frequency of the 11-Mbit/s SOQPSK signal)by setting the tuning parameter to k = 90. This places a copy of the too-narrow bandpass filterat the discrete-time frequency 90.5/360 cycles/sample as shown. But this is not enough. If the filteris properly designed (more on this below), it is possible to select the entire signal by creating abandpass 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 isstated as follows:

Key Concept 1: Because filtering is a linear operation, the signal producedby a filter that is the sum of the seven filters is equal to the signal producedby the sum the outputs of the seven individual filters.

2

LNB ADC r(nT )

➊ ➋ ➌

e−jΩ0n

y(nT )

↓D y(mDT )G(z)

1485.5 1496.5 1514.5 1435.0 1535.0

L-Band

frequency (MHz)

≈

450.5 461.5 479.5 400.0 500.0

LNB Output

frequency (MHz)

≈

frequency (cycles/sample)

90.5

360

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G(ejΩ)(k = 90)

G(ejΩ)(k = 92)

G(ejΩ)(k = 94)

G(ejΩ)(k = 96)

G(ejΩ)(k = 88)

G(ejΩ)(k = 86)

G(ejΩ)(k = 84)

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G(ejΩ), k = 101

❹

➊

➋

➌

❹

discrete-time channelizer

Figure 1: A version of the discrete-time channelizer from the companion paper [1]. A blockdiagram of the channelizer is shown at the top. The remaining diagrams are representations of thespectra at different points in the system.

3

r(nT )

H 0(z

M/D)

H 1(z

M/D)

H M−1(z

M/D)

...ge

nera

lized

com

mut

ator

ej2π(M−1)( kM + 1

2M )

yk(mDT )

e−j2π( kM/D

+ 12M/D )m

ej2π( kM + 1

2M )

phase shifting network

Figure 2: A polyphase filterbank realization of the channelizer in the dashed box in Figure 1.

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. Aremarkably 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 discrete-time channelizer. The polyphase filterbank version of the discrete-time channelizer was developedin [1] and is described by the block diagram of Figure 2. This system replaces the blocks in thedashed box of Figure 1. In Figure 2, the integer M is the ratio of sample rate to channel bandwidth(that is 1/(∆fT ) = M = 360 and that is why the first Nyquist zone at the input sample rate ispartitioned into 360 segments.) The downsample factor D determines the output sample rate (it is360/D in this example) and partly determines the properties of the subfilters in the filterbank. Togenerate the polyphase filterbank of Figure 2, 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 bandpassfilter shown in the examples of Figure 1. The filter H ′(z) is obtained from H(z) by changing thesign on every other filter coefficient. The bank of multipliers applied to the subfilter outputs arephase shifters whose phase shift is defined by M , D and k, the channel selection parameter. Thisbank of multipliers is identified by the dashed box labeled “phase shifting network” in Figure 2For a each value of 0 ≤ k < M = 360, the phase shifters cophase the components in the subfilteroutputs corresponding to the desired channel and cancel out the components corresponding to theother channels. When D < M not all of the channels wind up at baseband. The final multiplierperforms 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 fromleft 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 filterbankmust preserve all of the data from all M = 360 channels. Thus, we have the second key concept:

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Key Concept 2: The signal components from all M channels is simultane-ously available at the outputs of the M subfilters of the polyphase filterbankof Figure 2.

This concept was illustrated for the caseD = 90 in [1]. This illustration is repeated here in Figure 3(a). Stated in more general language, the generalized commutator aliases all of the channels to oneof M/D segments in the discrete-time frequency domain. The bandwidth of each segment isD∆F = D/M cycles/sample relative to the output sample rate. As a second example, the case forD = 15 is illustrated in Figure 3 (b). Here, all of the channels alias to one of M/D = 24 segmentsof bandwidth D∆F = D/M = 1/24 cycles/sample relative to the output sample rate. The channelswhose index k reduces to 0 modulo 24 alias to the segment centered at 0.5/24 cycles/sample relativeto the output sample rate; the channels whose index k reduces to 1 modulo 24 alias to the segmentcentered at 1.5/24 cycles/sample relative to the output sample rate; and so on. The imagery ofFigure 3 (b) suggests all of the signals possess the same bandwidth D∆F cycles/sample relativeto the output sample rate. The more realistic scenario, represented by the running example, isportrayed in Figure 3 (c). The SOQPSK signal centered at 1485.5 MHz aliases to the channelindexed by k = 90 (that is, the signal spectrum is centered at 90.5/360 cycles/sample relative to theinput sample rate — see Figure 1). Because k mod (M/D) = 90 mod 24 = 18, this signal aliasesto 18.5/24 cycles/sample relative to the output sample rate and has a bandwidth that spans 13 channelsegments as shown. Similarly, the PCM/FM signal centered at 1496.5 corresponds to the channelindexed by k = 101 (again, see Figure 1) and because 101 mod 24 = 5, the signal aliases to 5.5/24

cycles/sample relative to the output sample rate and has a bandwidth that spans 1 channel segment.The SOQPSK signal centered at 1514.5 MHz corresponds to the channel indexed by k = 119 andbecause 119 mod 24 = 23 the signal aliases to 23.5/24 cycles/sample relative to the output samplerate and has a bandwidth that spans just under 13 channel segments.

The main point is that copies of the bandpass filter may be produced in the polyphase filterbankrealization by only replicating the phase shifting networks. To create a bank of bandpass filterscentered at different frequencies, each phase shifting network uses a different value of k and all ofthe phase shifting networks process the same data. The overlap of the spectra in Figure 3 need notconcern us because the phase shifting networks eliminate the undesired signal components.

Combining Key Concept 1, Key Concept 2, and the bottom plot in Figure 1, we observe that asignal spanning 13 channel segments may be produced by applying 7 copies of the phase shiftingnetwork, each with a different value of k, to the subfilter outputs. This approach is illustrated inFigure 4. Here, the bank of phase shifters produce components of the SOQPSK signal centered at90.5/360 cycles/sample relative to the input sample rate (see the bottom plot of Figure 1). The endresult is the series of samples representing the I/Q baseband version of the 13 MHz of bandwidthcentered at 1485.5 MHz with a sample rate of 360/15 = 24 Msamples/s.

PROTOTYPE FILTER DESIGN

In its most basic form, the properties of the prototype lowpass filterH(ejΩ) satisfy those in Figure 5(a). Here, we have used the adjacent channel slot for the transition band. This is a reasonable

5

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ple

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freq

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ycle

s/sa

mpl

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k=0mod24

k=1mod24

k=11mod24

k=12mod24

k=23mod24

18.5

24=

−5.5 24

23.5

24=

−0.5 24

5.5

24

freq

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ycle

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360

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phase shift network (k = 96)

phase shift network (k = 94)

phase shift network (k = 92)

phase shift network (k = 90)

phase shift network (k = 88)

phase shift network (k = 86)

phase shift network (k = 84)

e−j2π 18.524 m

H 359(z

24)

H 1(z

24)

H 0(z

24)

. . .

Figure 4: Polyphase filterbank channelizer based on M = 360 and D = 15 and configured toselect a bandwidth of 13/360 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 telemetryapplications. 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 havea transition band that exhibits the symmetry illustrated in Figure 5 (b). The symmetry conditionsrequires that the sum of the transition band and a reversed copy of itself sum to a constant. (Theconstant is 1 here.) The formal definition of the symmetry is the following:

1 =∞∑

l=−∞

H(ej(Ω−2π2l∆F )

). (2)

The individual summands for l = −2,−1, 0, 1, 2 are plotted in Figure 5 (b). Note that the symme-try requirement means transition band of H(ejΩ) must pass through 0.5 at Ω = 2π∆F .

The filter response shown in Figure 8 of [1] approximately meets these requirements. The filterwas designed using the Parks-McClellan equiripple algorithm [2]. The filter design algorithmformed the basis for a design loop that iteratively adjusted the stop band to create a transition bandwhose magnitude was 0.5 at the desired frequency. (See Appendix A of [3] for a description ofthis approach to filter design applied to a slightly different application.) A filter designed using theParks-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 goodapproximation to the desired result.

7

0∆F

2

3∆F

2

1

2

1

A

frequency (cycles/sample)

pass band

transition band stop band

0

1

frequency (cycles/sample)

∆F 2∆F 3∆F 5∆F−5∆F −3∆F −∆F−4∆F −2∆F 4∆F

Hej(Ω−2π2∆F )

H

ejΩ

H

ej(Ω−2π4∆F )

H

ej(Ω+2π4∆F )

H

ej(Ω+2π2∆F )

(a)

(b)

Figure 5: An illustration of the design parameters for the prototype low-pass filter: (a) the pass-band, transition-band, and stop-band definitions; (b) an illustration of the transition-band symmetryrequired to produce a composite band-pass filter.

GENERALIZATIONS

The general form of the discrete-time channelizer based on a polyphase filterbank is illustrated inFigure 6. The only difference between this channelizer and the fixed-bandwidth channelizer ofFigure 2 is how the subfilter outputs are processed. The fixed-bandwidth channelizer applies asingle phase shifting network to the subfilter outputs whereas the variable-bandwidth channelizerapplies multiple phase shifting networks to the subfilter outputs. The center frequency is definedby the parameter k and the bandwidth is defined by the number of filter banks and their associatedindexes. Because the prototype filter has a bandwidth of ∆F = ∆fT cycles/sample at the inputsample rate (this corresponds to ∆f Hz), the discrete-time channelizer of Figure 6 is capable ofselecting bandwidths that are odd multiples of ∆f :

B = (2N − 1)∆f Hz N = 1, 2, . . . (3)

in continuous time and

BT = (2N − 1)∆F cycles/sample N = 1, 2, . . . (4)

in discrete time. To select a channel with bandwidth (2N − 1)∆F cycles/sample centered on thechannel indexed by k, the outputs of N phase shifting networks parameterized by the elements ofthe set K are summed. The N elements of the set K are

K =k − (N − 1), k − (N − 1) + 2, . . . , k + (N − 1)− 2, k + (N − 1)

(5)

8

r(nT )

H 0(z

M/D)

H 1(z

M/D)

H M−1(z

M/D)

...ge

nera

lized

com

mut

ator

yk(mDT )

e−j2π( kM/D

+ 12M/D )mN

pha

se sh

ift n

etw

orks

M inputs N outputs

Figure 6: The polyphase filterbank realization of the discrete-time channelizer capable of selectingchannels with variable bandwidth. This is a generalization of the block diagram of Figure 4.

where the special cases for N = 1, N = 2, and N = 3 are

K =k, N = 1

K =k − 1, k + 1

, N = 2

K =k − 2, k, k + 2

, N = 3.

(6)

This shows that the phase shifter arrangement depends on the parity1 of N . When N is odd, the setK includes the index k and this applies the bandpass filter centered at (k+0.5)/M as illustrated bythe example in Figure 7 (a). When N is even, the set K skips the index k and applies the bandpassfilters in the arrangement illustrated by the example in Figure 7 (b).

Because the downsample factor D must divide the parameter M [1], D cannot be selected arbi-trarily. For the running of example using a sample rate of 360 Msamples/s and ∆f = 1 MHz,we have M = 360. The possible values for D are the divisors of 360. These values, along withcorresponding output sample rate and usable RF bandwidth are listed in Table 1.

We are now in a position to describe how the variable-bandwidth discrete-time channelizer config-ures 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 con-trol.

• We assume the ADC sample rate 1/T and ∆f are fixed in advance. This defines the param-eters ∆F and M which, in turn, set the requirements for the prototype low-pass filter H(z).This meansH(z) is designed once by the receiver designer and is stored in the receiver. Thus

1The parity of an integer is its property of being even or odd.

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as the telemetry engineer changes the center frequency and bandwidth settings, the prototypelow-pass filter does not have to be recomputed.

With these assumptions in place, the receiver configuration proceeds as follows:

1. User input: The telemetry engineer selects the center frequency f0 (in Hz) and the bandwidthB (in Hz).

2. Calculate channel selection parameter k: After sampling the LNB output, express the de-sired center frequency as F0 = ∆F (k + 1/2). From this, the parameter k is computed using

k =F0

∆F− 1/2.

3. Calculate down sample factor D: The downsample factor may be selected in a number ofways. An obvious approach is to find the largest divisor of M for which M/D is greaterthan the normalized bandwidth BT . See Table 1 for an example for the case M = 360.

4. Set up phase shifting network: Express the normalized bandwidthBT (with units cycles/sample)as BT = (2N − 1)∆F and calculate N using

N =1

2

(BT

∆F+ 1

).

The parameter N indicates how many phase shifting networks are needed whereas the pa-rameters k and N determine which phase shifting networks are needed.

5. Configure the channelizer: Each new value of D requires a reconfiguration of the commu-tator, the spacing of the zeros in the subfilters of the polyphase filterbank, and the phaseshifting network. In an FPGA implementation, this constitutes a new “bit stream” load thatis 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 networkand the residual heterodyne. In a routine FPGA implementation, this can be accomplishedby changing a register value and does not require a new “bit stream” load.

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k∆F

(k + 2)∆F

(k + 1)∆F

(k + 3)∆F

(k + 4)∆F

(k + 5)∆F

(k + 6)∆F

(k − 1)∆F

(k − 2)∆F

(k − 3)∆F

(k − 4)∆F

(k − 5)∆F

kk−

2k−

4k

+2

k+

4

cente

rfr

equen

cy=

(k+

1 /2)∆

Fce

nte

rfr

equen

cy=

(k+

1 /2)∆

F

(k + 6)∆F

(k − 5)∆F

bandw

idth

=9∆

F

(k + 5)∆F

(k − 4)∆F

k∆F

(k + 2)∆F

(k + 1)∆F

(k + 3)∆F

(k + 4)∆F

(k + 5)∆F

(k + 6)∆F

(k − 1)∆F

(k − 2)∆F

(k − 3)∆F

(k − 4)∆F

(k − 5)∆F

cente

rfr

equen

cy=

(k+

1 /2)∆

Fce

nte

rfr

equen

cy=

(k+

1 /2)∆

F

(k + 5)∆F

(k − 4)∆F

(k − 3)∆F

(k + 4)∆F

bandw

idth

=7∆

Fk

+1

k+

3k−

3k−

1

(a)

(b)

Figu

re7:

Exa

mpl

esof

the

band

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filte

rar

rang

emen

tsto

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ndw

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=5,

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anex

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nN

.

11

Table 1: Possible values for the downsample factor D for the discrete-time channelizer based onthe polyphase filterbank shown in Figure 6 for an input sample rate of 360 Msamples/s and ∆f = 1MHz. These values defined M = 360.

maximumoutput passband

D sample rate bandwidth(Msamples/s) (MHz)

1 360 3602 180 1803 120 1204 90 905 72 726 60 608 45 459 40 40

10 36 3612 30 3015 20 2018 18 1820 15 1530 12 1236 10 1040 9 946 8 860 6 672 5 590 4 4

120 3 3180 2 2360 1 1

12

COMMENTS ON IMPLEMENTING THE PHASE SHIFTING NETWORK

The IFFT stuff goes here.

CONCLUSIONS

A variable-bandwidth version of the discrete-time channelizer developed in [1] as been described.The channelizer is based on an RF architecture that uses an LNB to translate the block of channelsin the desired frequency band (L-band, S-band, C-band, etc.) to a common frequency, and adiscrete-time architecture that uses an ADC coupled with a polyphase filterbank whose parametersare easily adjusted to select different bandwidths centered at all the center frequencies available inthe LNB output. The system is capable of selecting bandwidths that are odd multiples of 1 MHz.

REFERENCES

[1] 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 2011.

[2] A. Oppenheim and R. Schafer. Discrete-Time Signal Processing. Prentice-Hall, Upper SaddleRiver, NJ, 2009.

[3] M. Rice. Digital Communications: A Discrete-Time Approach. Pearson Prentice-Hall, UpperSaddle River, NJ, 2009.

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