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Many modern RF signal generators are equipped with a baseband signal processing board that includes a dual arbitrary waveform generator, which is used to drive an IQ modulator. These signal generators typically support a wide variety of modulation schemes implemented either as part of their firmware personalities¹ or via dedicated external software packages^2,3.

A high degree of support is available for cellular and wireless LAN based standards as a result of market demands. Such firmware and software personalities typically include the ability to fully encode the data stream, insert the necessary preambles, mid-ambles and postambles and apply the necessary pulse shaping.

By contrast, for niche or specialized applications encountered in such markets as aerospace or defense, the use of custom waveforms for device and system testing is commonly required.

Again, some vendors offer software based personalities to construct multitone, pulsed, and modulated pulsed and chirped signals. Advanced capability including baseband pre-distortion to improve image rejection, LO leakage and modulator frequency response are included⁴. However, such personalities are generally instrument specific. EDA tools, such as Agilent Technologies Inc.'s Advanced Design System, offer connected solutions such that a waveform may be generated in software and downloaded into the signal generator.

An alternative, when none of the above options meet the requirements is to design the waveform in customized software. Such an approach offers the waveform designer control on a complex sample-by-sample basis. Further, the instrument driver layer can be abstracted to enable signal generation across a variety of different instruments.

This paper examines several considerations for designing signals to be used with arbitrary waveform generators.

Waveforms can be divided into two categories, pulsed and continuous. With pulsed waveforms, phase continuity is generally not a consideration. However, with continuous waveforms, it is vital that phase continuity be preserved. If phase continuity is not maintained, then phase glitches, such as unintentional phase modulation, will arise, destroying the fidelity of the waveform.

To illustrate this effect, consider the generation of a simple sine wave of frequency f_o, comprising N samples at a sample rate of F_s, such that the sampling interval is T_s=1/F_s. A complete cycle of the sine wave is defined over the period 0≤t≤(N-1)T_s, such that the period is T=1/f_o=NT_s. It is important to note that the last time instant of the waveform is at (N-1)T_s and not NT_s.

Thus, for a sine wave of frequency f_o to be phase continuous, it must satisfy:

where m is an integer. In addition, due to Nyquist considerations, f_o<F_s /2, leading to the additional constraint that:

An N sample baseband multitone signal could be generated in the time domain by defining a fundamental frequency f_o where f_o=F_s /N. It would then be straightforward to develop a loop to calculate:

where S is the time domain multitone signal, A_m and Φ_m are the amplitude and phase of the m^th multitone respectively and n is an integer array defined from 0 to N-1, where N is the number of points in the waveform and the waveform period is NT_s.

Alternatively, the same waveform could be designed in the frequency domain and transformed into the time domain using the real part of the inverse Fast Fourier Transform (IFFT) such that:

where:

for N even with A_n and Φ_n are the amplitude and phase of the n^th multitone and Re denotes the real part of the signal (required due to numerical precision of PC-based FFT algorithms). A0 would represent the DC content of the waveform over its period and would generally be set to zero. The upper half of F_n is the complex conjugate of the lower half of F_n to provide a real waveform. Note that the case N even is generally used to obtain ease requirements of setting the sampling frequency.

The advantage of the FFT approach is that it avoids a looping construct in the code and it is conceptually simple to handle multitone waveforms that are not necessarily separated by f_o. For instance, consider the case where the test signal should be a multitone waveform, in which there is a single tone in each 1 MHz channel and the tones can be placed anywhere in the channel to a resolution of 100 kHz and the minimum over sampling ratio is four.

Figure 1 shows the VEE code to generate such a waveform, where Fval is the array containing the frequencies of the tones (Fval={0.1,1.4,2.7,3.5,4.9,5.2, 6.8,7.4,8.1,9.8}) and sigVal is an array of complex numbers each with unity magnitude and a random phase. Line 11 of the code generates the complex conjugate of sigVal for the upper half of the FFT.

For carrier modulated signals, the FFT technique can be further exploited by noting that the first half of the baseband frequency array corresponds to the upper sideband (USB) and the upper half of the baseband frequency array corresponds to the lower sideband (LSB), as illustrated in figure 3. If the upper half of the baseband frequency array is the complex conjugate of the lower half, the time domain signal is real and, thus, the output of an I/Q modulator is double sideband, suppressed carrier amplitude modulation. However, if the two halves of the frequency array are not equal, the IFFT is complex such that the real and imaginary parts of the IFFT are fed to the I and Q inputs of the I/Q modulator, respectively. In this case, the frequency components contained in F_n (1≤n≤(N-1)/2) form the upper sideband, while those in F_n ((N+1)/2≤n≤N-1) form the lower sideband. Note that F₀ represent the frequency component at the carrier. Due to the quadrature nature of signal generation, the maximum bandwidth of the signal is F_S.

Multitone signals can be used in noise power ratio (NPR) tests by removing the carrier and sometimes several of the tones around the carrier. Figure 4 shows the VEE code to generate 200 randomly phased multitones spaced at 100 kHz intervals either side of the carrier.

Figures 5 and 6 show the measured spectrum from the PSG signal generator. The component at the carrier is finite as a result of the LO leakage in the I/Q (caused by mixer LO leakage and I/Q offsets that drive the mixers).

If this level is unacceptable for the measurement application, there are a number of approaches to mitigate the LO leakage. In most modern signal generators it is possible to make I/Q adjustments and, thus, the LO leakage could be minimized by monitoring on the LO term on a frequency selective receiver (such as a spectrum analyzer) and adjusting the I/Q parameters under algorithmic control.

A second, more sophisticated approach would be to deliberately inject a component of equal amplitude and opposite phase to that of the LO leakage. However, this approach would require the repeated rebuilding and downloading of the waveform and, again, would require the use of a receiver.

The final and perhaps most expedient approach in ATE environments would be to simply shift the entire waveform such that the LO leakage fell on one of the multitones. If the LO is > 20 dB below one of the multitones, the worst case change in amplitude of this tone would be less than 1 dB. Alternatively, if the frequency resolution of the waveform was sufficient, the LO could be placed between two tones well away from the notch.

Most waveform generators require the waveform to be scaled for the DAC in either a signed or unsigned format. If the DAC or the I/Q modulator contains significant non-linearties, it can be desirable to include a scaling factor when digitizing the waveform to avoid driving either at full scale, particularly when generating signals for out-of-channel type measurements.

For the case of a D bit DAC, which uses unsigned integers, the waveform, negative full-scale output corresponds to 0, while positive full-scale output corresponds to 2^D-1. The level 2^D-1 corresponds to 0 volts after level shifting on the output board. For a system that uses signed integers, negative full scale output is at 2^D-1 and positive full scale output is given at 2^D-1-1, with 0 corresponding to 0 volts output. The code in figure 7 provides the scaling and digitizing functionality for the unsigned case. Where Tsig is the complex time domain waveform, D is the number of DAC bits and S represents the scaling factor (S=1 corresponds to full scale).

As a result of the sampled nature of the signal output by the DAC, images are generated centered on integer multiples of the sampling frequency. These images are removed using a low pass reconstruction filter as illustrated in figure 8.

Typically, a signal generator would offer the user a selection of two or three filter bandwidths to provide some flexibility in trading off waveform period and sampling rate against image rejection. Figure 9 shows the baseband output from an ESG-B for a 41 randomly phased multitone signal with a start tone frequency of 1 MHz and a tone spacing of 0.25 MHz with a clock rate of 40 MHz. The reconstruction filter is switched out of the path. The image components around the clock frequency are clearly visible. In addition the combined non-linearities of the DACs and I/Q modulators result in intermodulation components that fall both in and out of band. One can attempt to reduce this distortion empirically by applying a scaling factor to avoid driving the DACs at full scale.

In addition, the sample and hold action of the DAC gives rise to a sinc (x) frequency response of the form:

Thus, ignoring any roll-off in the signal path, a baseband over sampling ratio of 4 will result in the highest frequency component being attenuated by 0.91 dB. Obviously, the higher the over sampling ratio, the lower the roll-off effect from the DAC. If roll-off in the waveform cannot be tolerated, the signal should be convolved with a pre-emphasis filter, an example of which is discussed in “Generating Digital Modulation With the Agilent ESG-D Series Dual Arbitrary Waveform Generator Product Note,” Publication No. 5966-4097E (Agilent Technologies Inc., March 2002).

Over sampling DAC technology is now being used in some company's series' of signal generators, which alleviates the problems of image rejection and sinc (x) correction.

In this instance, the digital signal from the AWG, running at a maximum clock rate of 100 MHz is up-sampled to 400 MHz by the DACs. Figure 10 shows the baseband output from a PSG for 41 randomly phased multitone signals with a start tone frequency of 1 MHz and a tone spacing of 0.25 MHz with a clock rate of 40 MHz. The absence of signal images and signal roll-off are readily apparent.

In addition, the over sampling architecture provides the designer with greater flexibility in choosing sample rates since the constraint of adequate image rejection is removed. Note also the reduced distortion products as a result of the improved DAC performance (16 bit DACs).

The previous case study illustrated signal design in the frequency domain. Chirp signals are commonly encountered in radar and ranging applications, but may also be used as the stimulus signal for device characterization. Such signals are best designed in the frequency domain. A linear baseband chirp signal is given by:

where f_o is the start frequency, B is the chirp bandwidth and T is the period of the chirp signal. This expression can be resolved into its in-phase and quadrature components suitable for I/Q modulation as:

where Ω is the quadratic phase term, πp(B/T)t². The phase is the integral of frequency and, thus, it is possible to define the chirp in terms of its frequency versus time profile as illustrated in figure 11.

In this example, the chirp period is 0.2 ms, the chirp bandwidth is ±3 MHz relative to the LO and the sampling rate is 50 MHz. The phase versus time profile, phi, is obtained through numerical integration of the frequency ramp, DF.

Figure 12 shows the PSG measured output using a vector signal analyzer (VSA), to obtain both the spectral characteristics and demodulated FM waveform. The sharp transitions in the demodulated signal versus time are due to the phase discontinuities within a finite measurement bandwidth that arise between the frequency ramps.

Imperfections in the I/Q modulator, such as LO leakage, would manifest themselves as noise-like ripple on the frequency versus time measurement. In this instance, the raw I/Q modulator performance is sufficient to provide a clean chirp signal.

To further illustrate the power of the “frequency integration” technique, consider the case where the signal dwells at the start and stop frequencies for 2,000 samples (40 µms). The code in figure 11 is modified to resemble figure 13.

The measured output on the VSA from the PSG is shown in figure 14.

These two examples illustrate the ease with which such signals can be developed in the time domain. It would be straightforward to pulse the signal by suitably weighting the time domain profile that could include sample-by-sample control of the rise and fall times. The extinction ration would be limited by the LO leakage term. In signal generators equipped with a pulse modulator, it would be possible to mitigate the LO leakage from the I/Q modulator by pulsing the signal through this modulator.

This paper examined the design of waveforms for arbitrary signal generators and shown that with sufficient mathematical support, signals can be rapidly prototyped in software. Phase continuity requirements have been reviewed and a number of practical considerations have been highlighted. Signal design examples in both the frequency and time domains have been presented to illustrate.

1. “E4438C User Manual” (Agilent Technologies Inc.).

2. “E4438C Opt 417 Signal Studio for 802.11 WLAN Datasheet” (Agilent Technologies Inc.).

3. “WinIQ Sim,” (Rhode & Schwarz GmbH & Co.).

4. “E8267C Opt 420 Signal Studio for Pulse Building Datasheet,” (Agilent Technologies Inc.).

The author would like to thank Tim Masson and Colville Crooks, both of Agilent Technologies Inc. for helpful comments and review during the development of this paper for assistance in obtaining the measurement results.

Andy Street is a technical consultant with Agilent Technologies Inc. (www.agilent.com). He focuses on providing custom measurement solutions and test techniques for RF/microwave wireless applications.