We report first results of polyphase-coded incoherent scatter measurements at Millstone Hill. To our knowledge, these are the first incoherent scatter measurements with polyphase alternating codes of Markkanen et al. (2008) and optimal quadriphase sequences of Damtie et al. (2008). The results demonstrate that an arbitrary waveform generator recently installed at the Millstone Hill incoherent scatter radar, part of the National Science Foundation Geospace Facility operated by the Massachusetts Institute of Technology Haystack Observatory, is capable of reproducing the polyphase waveforms with an accuracy sufficient for incoherent scatter measurements. Polyphase codes will allow incoherent scatter radar experiments to be better optimized, because they provide a larger variety of code and code cycle lengths than the traditional binary codes.
 Incoherent scatter radars are high-power large-aperture radars that are used for detecting the weak radio wave scattering from thermal fluctuations in the ionospheric plasma. Pulse compression by means of binary phase coding is routinely applied in incoherent scatter measurements, often in the form of Barker codes [Barker, 1953] or alternating codes [Lehtinen and Häggström, 1987; Sulzer, 1993].
 Barker codes are mainly used in the ionospheric D-region, because coherence requirements set by the Barker code decoding can typically only be met there. At higher altitudes, the Barker codes can be used for subcoding individual bits of other modulations. Binary Barker codes are only known for code lengths of 2, 3, 4, 5, 7, 11, and 13 bits.
 Barker codes are conventionally decoded by means of matched filtering, which causes range sidelobes to the decoded signal. The definition of Barker codes requires that the amplitude of a range sidelobe of an Nb-bit code is at most a fraction 1/Nb of the mainlobe amplitude. Alternatively, the sidelobes can be suppressed by means of inverse filtering [Ruprecht, 1989; Lehtinen et al., 2004]. Because inverse filtering is applicable to almost any modulation, an extensive search for optimal binary codes for Nb=3,…,25 has been performed by Lehtinen et al. . Codes providing signal-to-noise ratio within 25% from optimal in inverse filter decoding were found for most of these code lengths.
 Alternating codes are designed for spectrally overspread targets, which makes them suitable for incoherent scatter measurements from the ionospheric E- and F-regions. Alternating codes consist of code cycles of Nc codes with Nb bits in each code. For the type 1 alternating codes of Lehtinen and Häggström , Nb=2m, where m is a positive integer and Nc=Nb for “weak” codes and Nc=2Nb for “strong” codes. Strong codes are typically used, because they allow uncoded bits to be transmitted without intervening gaps.
 Codes of an Nb-bit alternating code cycle can be truncated to any length shorter than Nb, but the length of the code cycle will still be Nc=2Nb for strong codes and Nc=Nb for weak codes. Type 2 [Sulzer, 1993] alternating codes provide code and code cycle lengths that are not restricted to powers of two, but these codes are only known for the lengths of 8, 9, 10, 11, 12 and 14 bits [Sulzer, 1993; Markkanen et al., 2008].
 An obvious possibility for finding more and better phase codes is to use more than two phases. A polyphase generalization of Barker codes was published already by Golomb and Scholtz , and new code lengths up to 77 bits have been published by several authors [Bömer and Antweiler, 1989; Friese, 1996; Brenner, 1998; Nunn and Coxson, 2009]. Similar generalizations are possible for type 1 alternating codes, [Markkanen et al., 2008] and for the codes optimized for inverse filtering [Damtie et al., 2008].
Damtie et al.  have evaluated individual phase codes in terms of the signal-to-noise ratio after inverse filter decoding, and performed an exhaustive search for optimal quadriphase codes for code lengths of Nb=2,…,22. The quadriphase coding provided codes with higher signal-to-noise ratio in inverse filter decoding than the binary coding in [Lehtinen et al., 2004] at several code lengths.
 The polyphase alternating codes of Markkanen et al.  have code lengths of Nb=pm or Nb=p−1 bits, where p is a prime and m is a positive integer. As the cycle length of strong codes is still Nc=2Nb, the polyphase generalization obviously provides a larger number of code cycle lengths when compared with the binary alternating codes.
 As an arbitrary waveform generator has recently been installed at the Millstone Hill incoherent scatter radar, part of the National Science Foundation (NSF) Geospace Facility operated by the Massachusetts Institute of Technology (MIT) Haystack Observatory, a possibility for testing the polyphase modulations in incoherent scatter measurements has now become available. We have performed a set of experiments in which both polyphase codes and closely similar binary codes were transmitted. We can thus investigate both the absolute performance of the modulations as well as compare the performance of polyphase codes with the traditional binary ones.
2 Phase Coding and Decoding
 Our data are two time series of voltage-level baseband signal samples: samples of the transmitted waveform and samples of the received signal both of which are recorded at regular time intervals tl=lΔt.
 Following the definitions of Lehtinen , the samples of the transmitted waveform are convolutions of a transmitter envelope env(t) and a receiver impulse response pr(t)
and the samples of the received signal are convolutions of a scattered signal zs(t) and the receiver impulse response
where we denote by‘∗’ the convolution operation.
 Similarly, following Lehtinen et al. , we define a discrete elementary pulse e
whose duration is Te=neΔt.
 The coding filter [e.g., Skolnik, 1990; Sulzer, 1989] of code number k in a cycle of Nb-bit codes can then written as
The code coefficients have two possible values ck,j=±1 in binary coding and np possible values ck,j= exp(i2πp/np),p=0,1,…,np−1 in polyphase coding with np phases. Actual transmission envelopes are obtained as convolutions of the coding filter and the elementary pulse
Codes in a code cycle are transmitted subsequently with an interpulse period Tp=npΔtbetween the codes. The final transmission envelope of a cycle of Nc codes will thus be
 Continuing from coding to decoding, we define a decoding filter
whose convolution with an elementary pulse filter q provides the final filter impulse response
The simple selection qn=en will be used in this paper, i.e. a similar boxcar shape is used for both the elementary pulse and its filter.
 We will use two different decoding filters: the matched filter
and the inverse filter
where Q(ω) and are the discrete Fourier transforms of the elementary pulse filter and the coding filter
The main difference between the matched and inverse filters is that a matched filter will produce range sidelobes, whereas the inverse filter will provide sidelobe-free decoding, but with a reduced signal-to-noise ratio. The quotient of the signal-to-noise ratios produced by the two filters can be calculated as the ratio
The ratio Rk has been used in Lehtinen et al.  and Damtie et al.  for evaluating individual phase codes.
 The matched filter in equation (10) suppresses sidelobes from each individual code. If the target is coherent in time, one can define an inverse filter for the whole code cycle,
This filter will produce sidelobes with individual codes, but the sidelobes will be canceled out when decoded signals from all codes are summed. If the inverse filter for a full code cycle reduces to the matched filter, the code cycle is a complementary code set [Golay, 1961]. The complementary codes would thus provide sidelobe-free decoding with an optimal signal-to-noise ratio but with the implicit restriction that the target must be coherent on time scales of at least two interpulse periods. The coherence requirement severely limits the use of complementary codes in incoherent scatter measurements.
 With alternating codes, the elementary pulse filter q is used as the receiver impulse response. One will then first calculate lagged products of the samples of the received signal, , j−l=τand decode these so-called lagged products. The decoding will produce a range profile of the scattering autocovariance function, which is called a lag profile at lag τ. Coefficients of the decoding filters are obtained as products of the code coefficients
Because the signal is correlated before decoding, alternating codes do not require target coherence on time scales longer than the elementary pulse.
 A special property of the alternating codes is that the coding filters formed from products of the code coefficients will produce complementary code sets at all nonzero lags shorter than the code length, i.e., at lags τ=n−m=1,2,…,Nb−1. Because the elementary pulse filter q will spread individual bits before calculating the products , this is not yet enough to guarantee sidelobe-free decoding, but the sidelobes will be suppressed only if the code cycle also fulfills the strong condition of Lehtinen and Häggström .
 Contributions from different ranges and Doppler frequencies in each sample of the received signal are conventionally calculated by means of ambiguity functions [e.g., Woodward, 1964]. In incoherent scatter lag profile measurements, contributions from the ranges S in the product can be calculated by means of the range ambiguity function of Lehtinen 
The range ambiguity functions of decoded lag profiles can obviously be calculated as weighted sums of range ambiguity functions of individual lagged products, with the weights given by the products of the code coefficients. The range ambiguity function can also be similarly calculated for codes decoded at the voltage level, if the appropriate decoding filters are substituted in equation (15).
 In addition to the full-integer lags at multiples of modulation bit length, one can oversample the signal and decode also noninteger lags of alternating codes using the decoding filters of the integer lags at either side of the noninteger lag value [Huuskonen et al., 1996]. Range ambiguity functions of the decoded noninteger lags will be narrower than those of the integer lags. The noninteger lags are important in several applications, but we will limit ourselves to the full-integer lags in order to maintain relative simplicity of this paper.
 If the code cycle is not an alternating code, one cannot suppress the range sidelobes from lag profiles by means of matched filtering. Almost arbitrary codes can be used for measuring lag profiles if lag profile inversion [Virtanen et al., 2008] is used instead of decoding. The technique will not be used in this paper, but we are interested in a code evaluation for lag profile inversion [Lehtinen et al., 2008]. The evaluation is very similar to the calculation of Rk in voltage-level decoding, but in this case the values of R(τ) will be calculated for full code cycles at all lags τ. Readers interested in the formulae for calculating R(τ) are referred to the original publication [Lehtinen et al., 2008].
 Four modulations, two binary-coded and two polyphase-coded, are investigated in detail in this paper. These are a 13-bit binary Barker code [Barker, 1953], a 25-bit strong binary alternating code (truncated from a cycle of 32-bit codes) [Lehtinen and Häggström, 1987], a 15-bit quadriphase code from Damtie et al. , and a 25-bit strong polyphase alternating code [Markkanen et al., 2008]. The strong polyphase alternating code was generated from a weak quinary code by repeating it twice and multiplying even bits of the second repetition with −1. The final polyphase alternating code thus had 10 different phases. Both alternating codes were randomized [Lehtinen et al., 1997] with the binary sequence −+−++−+−+−−−++−+−+−−++−−+.
 The Barker and quadriphase codes were both transmitted with 10 μs bit lengths and 3 ms interpulse periods. The interpulse period is somewhat too long for reasonable D-region autocovariance function measurements at nonzero lags, but this period was selected in order to avoid problems with aliased F-region echoes. The experiment still allows power profiles to be measured, which is well sufficient for our purposes. The alternating codes were transmitted with 20 μs bit lengths and 10 ms interpulse periods. Voltage-level samples of the transmitted and received waveforms were recorded from all experiments with a 500 kHz sample rate.
4 Inspection of Transmitted Waveform
 The recorded voltage-level samples of the transmitted waveform allow us to study the transmitted signal in detail. Examples of the samples are plotted in Figure 1 demonstrating that both pulse amplitudes and phases are reasonably stable. Some ripples at the bit edges and slight phase drift toward the ends of the longest pulses are visible. The phase drift measured from an individual pulse of the binary alternating code corresponds to an ion drift velocity of about 10 m/s. This error would be significant in several applications but can be easily accounted for in subsequent analysis steps if necessary.
 In order to get a wider view of phase stability, phases of individual samples of the transmitted waveform with small-amplitude samples cut off are plotted in Figure 2. Phases of the codes with 2, 4, and 10 phases are clearly concentrated in a corresponding number of separate groups, although the polyphase modulations seem to have a tendency to have a somewhat wider distribution for each nominal phase. The alternating codes have a minor phase drift during the measurement, whose source is unknown but will be investigated in future experiments. Anyhow, the drift is slow enough to be negligible in this case.
 The recorded samples of the transmitted waveform can be used as input for the code evaluations introduced in Lehtinen et al.  and Lehtinen et al. . The ratio of optimal posterior variance and that achieved in voltage-level inverse filtering or lag profile inversion, R, is tabulated for the different codes and lags in Table 1. Theoretical values of R are 0.952 for the binary Barker code, 0.933 for the quadriphase code, and 1 for all lags and voltage-level decoding of both alternating codes. Given that the largest deviation from an ideal value is only 0.012, we must conclude that all codes perform well in this test.
Table 1. Values of R and R(τ) for Different Codes and Code Cycles. The Values are Calculated Using Measured Samples of the Transmitted Waveform. The Zero-Lag Row Contains the Evaluation of Voltage-Level Decoding [Lehtinen et al., 2004]. Ideal Values Would be 0.952 for the Barker Code, 0.933 for the Quadriphase Code, and 1 for all Lags of Alternating Codes. Both Alternating Codes got Unit Value at all Nonzero Lags in this Evaluation
 Our primary interest in incoherent scatter measurements is the final shape of range ambiguity functions of decoded lag profiles, which can actually be measured using the recoded samples of the transmitted waveform. The original oversampled samples of the transmitted waveform were first filtered with the elementary pulse filter, i.e., a boxcar filter matched to the modulation bit length. Equation (15)) was then applied to the filtered oversampled signal in order to produce estimates of the true range ambiguity functions of individual undecoded pulses. Decoding filter coefficients were then calculated from equation (14) and final range ambiguity functions were calculated by means of appropriate multiplications and summations as explained in section (2). The code coefficients for decoding filter calculation were taken from the filtered and decimated samples of the transmitted waveform normalized to unit amplitude. Range ambiguity functions for different lags of the alternating codes are plotted in Figure 3.
 The samples of the transmitted waveform will unavoidably contain two random noise components which we cannot separate from each other: random errors in the transmitted waveform and thermal noise added on top of the true transmitted signal in the receiver. The range sidelobes should thus be reduced when increasing the number of code cycle repetitions, until systematic errors in the transmitted waveforms become dominant. For this reason, the plots contain curves for a single repetition and 1000 repetitions of the code cycle. If thermal noise from the receiver is the dominant noise contribution in the samples of the transmitted waveform, also short incoherent integrations will have range ambiguity functions similar to those integrated over 1000 repetitions in reality. On the other hand, if the noise is from random errors in the transmitted signal, short incoherent integrations will lead to larger sidelobes than longer ones.
 The plots demonstrate that both the binary and the polyphase alternating codes provide a 35 to 40 dB attenuation at unwanted ranges in long integrations. Given that the final range ambiguity function of the 25-bit code is 47 bits long at lag 1, and assuming that an equally strong scattered signal is received from all ranges, this would correspond to about 1% of total lag profile power to originate from range sidelobes in a worst-case scenario. Because the sidelobes have random-like phases, they will actually cancel each other out and the true leakage from range sidelobes will usually be significantly smaller. We will thus conclude that the ripples and phase drift visible in the samples of the transmitted waveform are not harmful for the sidelobe suppression in this case. If very short bits were used, the ripple could become problematic.
 The voltage-level pulse compression with a single code is simpler in the sense that the range ambiguity functions are identical at all lags. As the Barker codes should theoretically have at most unit-amplitude range sidelobes, one can calculate that sidelobes of decoded lag profiles from a 13-bit Barker code should be attenuated by 22.3 dB. Range ambiguity functions of the binary Barker code and the quadriphase code are plotted in Figure 4. Sidelobe levels of both codes are well below 20 dB in matched filter decoding, with the Barker code being reasonably close to the theoretical expectation. Although the quadriphase code was optimized solely for inverse filtering, it seems to perform well in matched filtering as well. Because range sidelobes will always be real and positive in this case, aliased signals from adjacent range gates may easily become significant. Matched filtering of modulations that are not perfect pulse-compression codes will always produce sidelobes, and the detected sidelobes must thus not be considered as defects of the transmitter.
5 Decoding Results
 Power profiles decoded by means of matched and inverse filters are plotted in Figures 5 and 6. Both decoding techniques reveal rather stable E- and F-regions. The F-region most probably looks weaker than it really is, because part of the power is destroyed in the voltage-level decoding of the spectrally overspread target.
 In order to estimate the background noise powers, the average power in range gates between 360 and 380 km was calculated for each integration period. It appeared that the interpulse period was not long enough to reach an altitude at which the incoherent scatter power could not be detected at all, but this range interval is used as the best alternative. Ratios of noise powers after matched and inverse filtering are plotted in Figure 7. The ratio is obviously in line with both the purely theoretical prediction and the one based on the measured samples of the transmitted waveform.
 Finally, incoherent scatter autocovariance functions measured with the binary and polyphase alternating codes are plotted in Figures 8 and 9. We do not have any special evaluation for these results, but neither of them contains artifacts that would give a reason to suspect errors in coding or decoding. The results from the two codes are clearly different at just below 100 km in range, but this is to be expected, because the two measurements were not simultaneous.
 As the binary coding techniques applied in a present-day incoherent scatter radar are already very good, the polyphase codes will not be able to provide dramatic improvements to the accuracy of incoherent scatter measurements. Their value is in the significantly enlarged number of different code and code cycle lengths, which will allow experiments to be better optimized according to specific needs.
 Signal processing of polyphase codes uses the same mathematical tools that are used in binary coding techniques. However, signal processing software designed for the binary codes may not be directly applicable to the polyphase codes, because the coefficients of the decoding filters will be complex, in contrast to the purely real filter coefficients of binary codes.
 Polyphase codes are a natural extension of incoherent scatter modulation techniques provided by the modern arbitrary waveform generators. As amplitude modulation is equally achievable with the same devices, and so-called perfect pulse-compression codes using amplitude modulation are known [Lehtinen et al., 2009], a natural next step will also be to investigate amplitude modulation. As binary phase coding is used in the codes of Lehtinen et al. , the decoding filter coefficients will be real again.
 In addition to variances of decoded lag profiles, mutual covariances of the decoded lagged products are important in incoherent scatter experiments. These covariances depend on both the applied modulation and decoding, as well as signal-to-noise ratio of the incoherent scatter signal.
 Polyphase alternating codes share the same properties with binary alternating codes [Markkanen et al., 2008]. All kinds of alternating codes will thus provide uncorrelated lag profiles in very low signal-to-noise conditions [Lehtinen and Häggström, 1987], whereas randomized codes must be used in order to suppress unwanted correlations in high signal-to-noise conditions [Lehtinen et al., 1997].
 Regarding voltage-level decoding, we do not have any reason to assume significant differences in between covariance structures of equally good binary and polyphase codes. However, perfect pulse-compression codes (R=1) provide uncorrelated decoding results [Lehtinen et al., 2009, section 6] when background noise is white noise—which is usually a good approximation in voltage-level decoding. One could thus guess that codes with higher R might have smaller correlations in some sense, but we do not have a proof for this.
 We have performed the first polyphase-coded incoherent scatter measurements at the Millstone Hill incoherent scatter radar, part of the NSF Geospace Facility operated by the MIT Haystack Observatory. The polyphase codes were shown to perform sufficiently well to be adopted in routine incoherent scatter measurements. The signal-to-noise ratio in voltage-level inverse filtering was found to be within one percent from theoretical expectations, and a better than 30 dB range sidelobe attenuation was achieved with alternating codes.
 This work has been funded by the Academy of Finland (application number 250215, Finnish Programme for Centres of Excellence in Research 2012-2017, and application 250252, Measurement Techniques for Multi-Static Incoherent Scatter Radars). Radar observations and analysis at Millstone Hill are supported under Cooperative Agreement ATM-0733510 with the Massachusetts Institute of Technology by the U.S. National Science Foundation.