DISCRETE-TIME CHANNELIZERS FOR AERONAUTICALTELEMETRY: PART I — FIXED BANDWIDTH

Brian Swenson, Michael RiceBrigham Young University

Provo, Utah, USA

ABSTRACT

A discrete-time channelizer based on a polyphase filter bank and suitable for use with aeronau-tical telemetry is described. The discrete-time approach reduces the number of continuous-timecomponents — and the need to maintain and calibrate these components. Channel selection isaccomplished by changing a single parameter that defines a small part of the system.

INTRODUCTION

A channelizer is a device that selects a channel where by “channel” we mean all of the energyin a contiguous frequency range centered at a specified frequency. The center frequency is thefrequency assigned by a range frequency manager to accommodate an RF telemetry signal. Thechannel bandwidth is determined by the properties of the RF telemetry signal, most notably thebit rate and the modulation type (e.g., PCM/FM, SOQPSK, ARTM CPM). The selection processis performed by attenuating the signal components outside the frequency range defined by thechannel. Fundamental to this process is the need for a frequency plan. In aeronautical telemetry,the frequency plan is defined in Chapter 2 and Appendix A of the IRIG 106 standard [1].

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 frequency translation (or heterodyne1 operation). A

1The term heterodyne is derived from the Greek heteros (ετερoς) meaning “other” and dynamis (δυναµις) meaning“force.” It was first coined by Reginald Fessenden in a 1901 patent application describing the application of the well-known property of a “beat frequency” — the difference frequency resulting from the simultaneous presence of twoaudio tones — to the detection of radio frequency signals. The original concept envisioned the transmission of two RFsignals each received with its own antenna and combined at the detector. Later, one of the two RF transmitter-antennasystems was replaced by an RF source generated at the detector (a “local” oscillator) to produce the version of theheterodyne receiver described in most introductory texts on communication systems today. Fessenden was awardedU.S. Patent 706,740 issued on 12 August 1902.

1

BPF ADC

coarsely tunable filter

LO1

BPF

IF1 center frequency

IF1

LO2

IF2

BPF 1

BPF 2

BPF N

IF Switch (IF bandwidth select)

to discrete-time demodulator

RF input (from LNA)

SAW Filterbank

...

IF Switch (IF bandwidth select)

Figure 1: A representative example of a dual-conversion channelizer based on the superheterodyneconcept. Because the focus is on the filtering and frequency translation operations associated withchannelization, signal level adjustments (amplifiers) are omitted.

typical example using dual conversion is illustrated in Figure 1. Channel selection is performedby adjusting the pass band of a coarsely tunable bandpass filter (whose primary purpose is imagerejection) and the frequency of the first local oscillator LO1. The result is that a portion of the RFspectrum is centered at in intermediate frequency IF1. Undesired components are eliminated by asecond bandpass filter centered at IF1 and the filter output is translated to the second intermediatefrequency IF2 by the second local oscillator LO2. (In some designs, the oscillation frequency ofLO2 is adjustable, in other designs it is fixed.) At this point, the signal is applied to a bank of SAWbandpass filters. Each filter in the filterbank has a different bandwidth. The telemetry engineer’schoice of IF bandwidth sets the switch position which, in turn, selects the filter used for the finalfiltering stage. At this point, the desired center frequency is centered at IF2 (typically 70 MHz). Inrecognition of the reality that most modern telemetry demodulators use discrete-time processing,an analog-to-digital converter (ADC) is included as the final step. It needs to be pointed out that inaddition to limiting the noise and adjacent signal components, the final IF filter also serves as theanti-aliasing filter for the ADC.

There are several challenges associated with this approach. Among the most important are thefollowing

1. A large number of continuous-time (analog) components that must be maintained and cali-brated.

2. Managing the accumulation of distortions (e.g., phase noise) introduced by the continuous-time components is often difficult.

3. The available IF bandwidths are determined by the bandwidths of the filters in the SAWfilterbank. Increasing the number of options available to the user increases the complexityof the design. This, in turn, exacerbates the first problem.

In this paper we describe an alternate approach, based on discrete-time processing, that addressesmany of these challenges. By moving the ADC closer to the antenna feed, continuous-time com-

2

L-Band LNB

UL-Band LNB

S-Band LNB

US-Band LNB

C-Band LNB

ADC to discrete-time demodulator

RF input (from LNA)

discrete-time channelizer

Figure 2: The basic architecture of a feasible discrete-time channelizer.

ponents may be eliminated. The flexibility of discrete-time processing can be exploited to improvethe options for IF bandwidth. These advantages come with a price: the sample rate must be highenough to capture the entire band. These challenges are addressed by using a polyphase filterbankto exchange clock rate for circuit area.

THE BASIC ARCHITECTURE OF THE DISCRETE-TIME CHANNELIZER

The ideal discrete-time channelizer samples an appropriately scaled signal output from the antennafeed. Even if ADCs could be clocked high enough to make this a feasible option for L- and S-bands, the current crop of discrete-time processors are incapable of handling (in real time) thetorrential flood of ADC output samples. Consequently, a hybrid approach is required. An exampleis illustrated in Figure 2. Here, a bank of low-noise block downconverters are used to translateeach frequency band to a common lower frequency. The ADC samples the LNB output. Thesamples are processed by a discrete-time channelizer (described in the next section) in preparationfor demodulation.

At first glance, this system may not to be much an improvement over the system of Figure 1. Butthis is not necessarily the case. The LNBs do not contain any user tunable elements. Consequently,its filters and mixers can be optimized for their respective frequency bands.

The discrete-time channelizer is described in the next section. The channelizer is designed to select1 MHz-wide portions of the input spectrum (1 MHz is the minimum channel spacing defined inIRIG 106). The variable bandwidth case is described in the companion paper [2].

3

THE MATHEMATICS OF THE DISCRETE-TIME CHANNELIZER

The channelizer problem for aeronautical telemetry is illustrated2 in Figure 3. The band is dividedinto channels of bandwidth ∆f = 1 MHz, the minimum channel spacing defined by IRIG 106.Associated with each channel is a center frequency. For the purposes of the channelizer, the centerfrequency at the LNB output is parameterized in terms of the channel separation ∆f using

f0 = k∆f + 0.5 (MHz) (1)

for some integer k. After sampling at 1/T samples/s, the frequency separation in the discrete-timefrequency domain is ∆F = ∆fT cycles/sample and the center frequency is

F0 = f0T = k∆F + 0.5× 106T cycles/sample. (2)

To make the polyphase filterbank work, we require the ratio of sample rate 1/T to frequencyseparation ∆f to be an integer. That is

M =1

∆fT=

1

∆Fsamples/cycle (3)

for some integer M . Consequently, the center frequency may be expressed as

F0 =k

M+

1

2Mcycles/sample. (4)

The development begins with the discrete-time system illustrated in Figure 4 (a) and its correspond-ing frequency-domain interpretation in Figure 5. The received signal comprises various real-valuedband-pass signals illustrated by R(ejΩ) in the figure. The channelizer of Figure 4 (a) isolates thereal-valued band-pass signal centered at Ω0 = 2πF0 rads/sample using the heterodyne/low-pass-filter operation shown. The heterodyne operation, achieved using the multiplier and samples ofa complex-exponential oscillating at Ω0 rads/sample, moves the desired center frequency to I/Qbaseband where the low-pass filter, with z-transform

H(z) =∑n

h(n)z−n (5)

isolates the I/Q baseband equivalent of the desired signal as illustrated in Figure 5 (a). This ap-proach, motivated by the superheterodyne channelizers (receivers) in common use, “moves thedata to the filter.”

The difficulty with this approach is the fact that the output is massively oversampled. That this is socan be deduced from the fact that at the input, the sample rate must be large enough to adequately

2The experienced reader will immediately notice a glaring problem with the spectra of Figure 3: all of the channelshave the same bandwidth and that this common bandwidth is the minimum channel spacing allowed in IRIG 106. Thereality is that the frequency plan allows variable-bandwidth channels where the bandwidth is allocated in ∆f = 1MHz increments. This paper focuses on the unrealistic case (all frequency allocations have the same bandwidth) tointroduce the concept. The realistic case of variable-bandwidth channels is described in the companion paper [2].

4

LNB ADC r(t)

r(nT )r(t)

f0∆f

L-band 1435 – 1535 MHz

R(f)

f

∆f

LNB output 400 – 500 MHz

f

R(f)

f0

Nyquist Zone

R(ejΩ)

F∆F F0

➊ ➋ ➌

➊

➋

➌

Figure 3: The spectra of the signals requiring channelization. For the purposes of illustration,the RF spectrum is assumed to be L-band (1435 – 1535 MHz). The same basic approach (withmodifications to the LNB output frequency range) works for Upper L-band (1710 – 1850 MHz),lower S-band (2200 – 2290 MHz), upper S-band (2360 – 2390) or C-band allocations (4400 – 4900MHz and 5091 – 5150 MHz).

5

represent approximately 100 MHz of bandwidth. The bandwidth of the low-pass filter output is 1MHz. For this reason, a downsample operation is added to the system as shown in Figure 4 (b).The downsample factor D is defined by

D =input sample rateoutput sample rate

. (6)

The downsample operation retains every D-th sample and, in effect, changes the sample timefrom T s/sample to DT s/sample. In recognition of the fact that the time axis has changed, theoutput in Figure 4 (b) uses the time index m in place of n (the time index used for the input).The downsample operation is the key that allows the efficiencies of the polyphase filterbank to beapplied. Note that D must divide M for the polyphase decomposition to work.

The first step in development of the polyphase filter bank is to transpose the heterodyne and filteringoperations. This is accomplished by re-writing the low-pass filter output of Figure 4 (a) as

y(n) =[x(n)e−jΩ0n

]∗ h(n) =

∑l

h(l)x(n− l)e−jΩ0(n−l)

=

[∑l

h(l)ejΩ0l︸ ︷︷ ︸g(l)

x(n− l)]e−jΩ0n =

[∑l

g(l)x(n− l)]e−jΩ0n

= [g(n) ∗ x(n)] e−jΩ0n. (7)

In this equation, the filter g(n) is obtained from the low-pass filter h(n) by heterodyning the filtercoefficients. This operation converts the low-pass filter h(n) to a bandpass filter g(n) centeredat Ω0. The resulting approach, illustrated in Figure 4 (c), uses the bandpass filter centered at Ω0

rads/sample to isolate the signal of interest: see Figure 5 (b). The band-pass filter output is movedto I/Q baseband using the heterodyne operation represented by the multiplication by the complex-exponential. In contrast to the approach shown in Figure 4 (a), this approach “moves the filter tothe data” and is the discrete-time equivalent of the tuned radio receivers commonly used in theearly days of broadcast radio.

The z-transform of the bandpass filter g(n) may be written as

G(z) = g(0) +g(M)z−M +g(2M)z−2M + · · ·+g(1)z−1 +g(M + 1)z−M+1 +g(2M + 1)z−2M+1 + · · ·...

+g(M − 1)zM−1 +g(2M − 1)z−(2M−1) +g(3M − 1)z−(3M−1) + · · ·

(8)

This is an M -stage polyphase decomposition of filter. Each row represents a subfilter in the corre-sponding polyphase filterbank. Using the relationship g(n) = h(n)ejΩ0n, the z-transform may beexpressed as

G(z) = h(0) +h(M)ejMΩ0z−M + · · ·+h(1)ejΩ0z−1 +h(M + 1)ej(M+1)Ω0z−M+1 + · · ·...

+h(M − 1)ej(M−1)Ω0zM−1 +h(2M − 1)ej(2M−1)Ω0z−(2M−1) + · · ·

(9)

6

H(z)r(nT )

e−jΩ0n

y(nT )

H(z)r(nT )

e−jΩ0n

y(nT )

↓D y(mDT )

r(nT )

e−jΩ0n

y(nT )

↓D y(mDT )G(z)

(a)

(b)

(c)

➊ ➋ ➌

➊ ➋ ➌

➊ ➋

Figure 4: Discrete-time processing of bandpass signals: (a) moving the data to the filter; (b) movingthe data to the filter followed by a downsample operation; (c) moving the filter to the data.

7

R(e

jΩ)

R(e

jΩ)

Ω

G(e

jΩ)

H(e

jΩ)

DTF

T at

➊

DTF

T at

➋

DTF

T at

➌

DTF

T at

➊

DTF

T at

➋

DTF

T at

➌

π−π

0Ω

0−Ω

0

π−π

0

π−π

0

π−π

0

π−π

0Ω

0−Ω

0

π−π

0

π−π

0

π−π

0

Ω0

Ω Ω Ω

Ω Ω Ω Ω

(a)

(b)

Figu

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).

8

G(z) =[h(0) + h(M)ejMΩ0z−M + h(2M)ej2MΩ0z−2M + · · ·

]+ ejΩ0z−1

[h(1) + h(M + 1)ejMΩ0z−M + h(2M + 1)ej2MΩ0z−2M + · · ·

]+ · · ·

+ ej(M−1)Ω0z−(M−1)[h(M − 1) + h(2M − 1)ejMΩ0z−M + h(3M − 1)ej2MΩ0z−2M + · · ·

](10)

At this point we examine the complex exponential terms inside the square brackets. Each term isof the form expjqMΩ0 for integers q = 0, 1, 2, . . .. Using the relationship (4) we see that

expjqMΩ0 = exp

jqM

(2πk

M+

2π

2M

)=

:1

exp jq2πk exp jqπ = (−1)q. (11)

This shows that the complex exponential terms inside the square brackets may be removed byalternating the signs on the terms in the row:

G(z) =[h(0)− h(M)z−M + h(2M)z−2M − · · ·

]︸ ︷︷ ︸H′0(zM )

+ejΩ0z−1[h(1)− h(M + 1)z−M + h(2M + 1)z−2M − · · ·

]︸ ︷︷ ︸H′1(zM )

...+ej(M−1)Ω0z−(M−1)

[h(M − 1)− h(2M − 1)z−M + h(3M − 1)z−2M − · · ·

]︸ ︷︷ ︸H′M−1(zM )

(12)This can be expressed in the more compact form

G(z) = H ′0(zM) + ejΩ0z−1H ′1(zM) + · · ·+ ej(M−1)Ω0z−(M−1)H ′M−1(zM). (13)

Using the Noble identities [3], the downsample operation at the output side of the block diagram inFigure 4 (c) may be moved to the input side. The resulting system is illustrated in Figure 6. Notethat the subscript “k” has been added to the output to emphasize the fact that this system selectsthe channel centered at the frequency indexed by k [see Equation (4)]. The important observationshere are the following:

1. The polyphase filterbank system of Figure 6 is equivalent to the discrete-time systems ofFigure 4 (b) and (c).

2. Each subfilter in the filterbank consists of real-valued coefficients — see Equation (12). Thisis a consequence of the relationship (11) that follows directly from the requirement that thefrequency separation divides the sample rate — see Equation (3).

3. Each subfilter operates at the output sample rate, not the input sample rate. The outputsample rate is 1/D times the input sample rate. Because the polyphase filterbank performsparallel computations at the low clock rate, the polyphase filterbank realization is a form ofparallel processing that exchanges clock rate for area.

9

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( k

M + 12M )

ej2π(M−1)( kM + 1

2M )

yk(mDT )

e−j2π( kM/D

+ 12M/D )m

Figure 6: A polyphase filterbank realization of the channelizer illustrated in Figure 4 (c).

4. Channel selection performed by changing the tuning parameter k. The tuning parameter konly shows up in the phase shifters applied to the subfilter outputs and the multiplier at theright-hand side of the figure. The means that channel selection is not performed until thebank of phase shifters is applied. Consequently, all of the M = 360 channels are simulta-neously available at the outputs of the subfilters. Furthermore, the multiple channels may beextracted from the input by replicating the bank of phase shifters, the adder, and the mul-tiplier — the filterbank (the resource-intensive component) does not need to be replicated.This feature will be exploited to realize a variable bandwidth channelizer in the companionpaper [2].

A DESIGN EXAMPLE

As an example, suppose the sample rate is 1/T = 360 Msamples/s and the desired output samplerate is 4 Msamples/s. The relations (3) and (6) give

M = 360 D = 90 ∆F =1

360

M

D= 4. (14)

The discrete-time prototype filter h(n) is designed at the high sample rate. The filter specificationsare derived from the functionality required by the system of Figure 4. The filter should be a lowpass filter that passes a channel of width ∆F cycles/sample (extending from −∆F/2 to ∆F/2cycles/sample) and attenuates the remaining channels. Provision for a transition band is required.The design specifications illustrated in Figure 7 meet these requirements. Here we have used thewidth-∆F channel adjacent to the passband as the transition band because it is rare (in aeronauticaltelemetry applications) that a user is placed in the channel immediately adjacent to the band-edge.A narrower transition band can be used, but this is achieved at the expense of filter length. The

10

0 ∆F

2

3∆F

2

1

2

1

A

frequency (cycles/sample)

pass band

transition band stop band

Figure 7: The design specifications for the discrete-time prototype filter h(n) of Figure 4.

filter used in this example was designed using the Parks-McClellan equi-ripple design procedure[4] with a target of 60 dB stop band attenuation. The shortest filter that meets these specificationsand whose length is a multiple of M = 360 is the length-1440 filter shown in Figure 8. Thepolyphase filterbank based on this filter consists of 360 length-4 subfilters.3 Even though the inputsample rate is 360 Msamples/s, all of the subfilters operate at 4 Msamples/s.

The system of Figure 6 has an interesting interpretation. The input commutator downsamples theinput data by D = 90. Consequently, all of the M = 360 channels at the input (high) sample ratealias to one of M/D = 4 positions in the at the output (low) sample rate. This is illustrated inFigure 9. At the output sample rate, the channel spacing is D∆F = D/M = 1/4. As the channelcenters are offset from the spacing by ∆F/2 (or by D∆F/2 = 1/8 at the output sample rate), thechannel centers are the odd multiples of 1/8 as shown. Each of the M/D = 4 positions, containcontributions from D = 90 of the M = 360 channels.

Each of the M = 360 subfilters is an all-pass filter. (The only difference between the frequencyresponses is the phase — hence the name “polyphase” filterbank.) Each subfilter imparts a differentphase slope on its input. The role of the complex-valued multipliers at the subfilter outputs isto apply additional phase shifts the subfilter outputs. The phase shifters alter the phase of thesubfilter outputs so that the desired signal components add constructively while the undesiredsignal components cancel. As a result, the signal at the output of the adder only contains the desiredchannel. The multiplier after the adder is a residual heterodyne operation required to translate thedesired signal from its center frequency (an odd multiple of 1/8) to baseband.

For example, the channel centered at F0 = 90.5/360 is the channel parameterized by k = 90 [seeEquation (4)]. Because 90 mod 4 = 2, this channel aliases to −3/8 cycles/sample. In addition, allof the other channels alias to this or one of the other three positions. The phase shifters co-phasethe components of the desired channel and phase-shift the other signal components so that the sumcontains only the desired channel. At the output of the summer, the desired channel is centered at−3/8 cycles/sample. The heterodyne translates this signal to baseband.

3Each subfilter is technically a length-16 filter with four non-zero coefficients: every 4-th coefficient. This is whatthe z4 means in H ′

m(z4).

11

−600 −400 −200 0 200 400 600−2

0

2

4

6x 10

−3

sample index n

h(n

)

−80

−60

−40

−20

0

frequency (cycles/sample)

| H(e

jΩ)|

(dB

)

− 2

360− 1 . 5

360− 1

360− 0 . 5

3600

0 . 5

360

1

360

1 . 5

360

2

360

Figure 8: The prototype filter used for the design example: (top) the impulse response; (bottom)the frequency response. Note that the filter impulse response (top plot) is discrete — the continuousline connecting the samples is used only for clarity.

0

90 channels indexed by k mod 4 = 0

alias to ⅛ cycles/sample

90 channels indexed by k mod 4 = 1

alias to ⅜ cycles/sample

90 channels indexed by k mod 4 = 2

alias to ⅝ cycles/sample

90 channels indexed by k mod 4 = 3

alias to ⅞ cycles/sample

−1

4−1

2−3

8−1

8

1

8

3

8

1

4

1

2

cycles/sample

Figure 9: A discrete-time frequency domain representation of data at the output of the filterbankshown in Figure 6 for the case M/D = 4.

12

As the final step in our example, we relate this system to L-band operation. Suppose the LNBtranslates L-band (1435 – 1535 MHz) to the frequency range 400 – 500 MHz and that the desiredchannel is centered at 1485.5 MHz. Assuming that the relationship between the L-band frequenciesand the LNB output are those shown in Figure 3, the relationship between the center frequenciesfor the RF, LNB output, and ADC outputs are4

1435.5 MHz → 400.5 MHz → 40.5

360cycles/sample

1436.5 MHz → 401.5 MHz → 41.5

360cycles/sample

...

1485.5 MHz → 450.5 MHz → 90.5

360cycles/sample

...

1534.5 MHz → 499.5 MHz → 139.5

360cycles/sample

The desired channel is centered at F0 = 90.5/360 = 90/360 + 0.5/360 cycles/sample. Conse-quently, the channel centered at 1485.5 MHz is selected when the tuning parameter of Equation (4)is set to k = 90. Because k mod M/D = 90 mod 4 = 2, the channel corresponding to 1485.5 MHzaliases to −3/8 cycles/sample, the left-most position in Figure 9. For k = 90 the residual hetero-dyne reduces to e−j2π(5/8)m, and this operation moves the signal centered at −3/8 cycles/sample tobaseband.

A different channel is selected by changing k. For example, the channel centered at 1492.5 MHzis selected by using k = 97. This changes the phase shifters applied at the subfilter outputs andchanges the frequency of the residual heterodyne operation to e−j2π(3/8)m. Because 97 mod 4 = 1,the downsampling operation aliases this channel to 3/8 cycles/sample and the residual heterodyneuses the correct frequency. The main point here is that changing the channel selected by thechannelizer only involves a change to the tuning parameter k, and the only circuit parameters thatdepend on k are the phase shifters and the residual heterodyne operation.

CONCLUSION

A discrete-time channelizer based on a polyphase filter bank and suitable for use with aeronau-tical telemetry is described. The discrete-time approach reduces the number of continuous-timecomponents — and the need to maintain and calibrate these components. Channel selection isaccomplished by changing a single parameter that defines a small part of the system. The rest of

4Most LNBs use high-side mixing that swaps the negative frequency and positive frequency components at the LNBoutput. (This is sometimes called “spectral inversion.” Though not a technically accurate description of the effect, theterminology persists.) With high-side mixing, the highest frequency channel (centered at 1535.5 MHz) shows up at thelow end of the LNB output (at 400.5 MHz) and the spectrum is “reversed.” In this example negative/positive frequencyswapping is not assumed, as would be the case with low-side mixing. In the case where such swapping occurs, thechanges required to the channel indexing are straight-forward.

13

the system remains unchanged. The primary limitation of this design is the requirement that allchannels be the same bandwidth, the minimum bandwidth defined in IRIG 106. Extensions thataddress this issue are described in the companion paper [2].

REFERENCES

[1] Secretariat, Range Commanders Council, White Sands Missile Range, New Mex-ico. IRIG Standard 106-00: Telemetry Standards, 2009. (Available on-line athttp://www.irig106.org/docs/106-09).

[2] B. Swenson and M. Rice. Discrete-time channelizers for aeronautical telemetry: Part II —variable bandwidth. In Proceedings of the International Telemetering Conference, Las Vegas,NV, October 2011.

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

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

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