Dimensión Cognitiva

After the PA characteristics, the next most important cartesian feedback parameter is the open loop frequency response of the system. The open loop frequency response governs the degree and bandwidth of the distortion reduction [42]. The higher the loop gain the greater the reduction. Traditional frequency response techniques can be applied to the cartesian feedback loop and offer a useful simplified starting point for further analysis. An equation to illustrate this for a cartesian feedback loop (as obtained from figure 3.7) can be written with all variables being complex as,

(3.1)

where Y(t) represents the transmitter output and X(t) represents the input signal. The open-loop gain G_A, is comprised of all the forward gains in the cartesian feedback system i.e (as shown in figure 3.7) the baseband amplifiers and filters (G(s)), the modulator, driver, and RF power amplifier (gejδ), and the transmission delay (ejωτ). d_A(t) models the distortion introduced by all of these forward gain components. The feedback transfer function H_fwhich is comprised of the RF directional coupler and demodulator, also has an associated distortion component d_f (t). The expression highlights how d_A(t) which includes the RF amplifier non-linearity, is approximately reduced by the amount of loop

Y t( ) ₁GA_GX t( ) AHf + --- dA( )t 1+G_AH_f --- d₁f( )t_GGAHf AHf + ---, – + = G_A H_f d_A(t) d_f(t) X(t) Y(t)

Figure 3.7: Complex baseband representation of cartesian feedback loop modelling gains and distortion. Bold lines signify complex quantities i.e. two lines. Forward gain, G_A, is comprised of the gain in the baseband amplifiers and filters G(s), the gain and RF phase rotation of the RF amplifier and upconvert chain, gejδ , and a delay, ejωτ.

G_A = G(s) g ejδ ejωτ

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gain G_AH_f. A loop gain of 35dB, for example, will reduce output intermodulation products by approximately 35dB. The loop still however remains sensitive to distortion generated in the feedback path (d_f (t)) which is not changed by the amount of loop gain. This highlights the need for the feedback gathering components to be highly linear and also of low noise[57]. All of the distortion quantities except that produced by the amplifier are generally small compared to the output signal and consist of terms which are either constant (such as noise power level, DC offset or carrier leak) or are signal dependent such as the amplifier intermodulation distortion. This section only considers the effects of amplifier distortion which can be reduced by the action of the feedback loop (i.e d_f (t) is assumed to be 0).

The modulation bandwidths discussed in this thesis are narrowband (10’s kHz) relative to the RF component bandwidths in the loop (10’s MHz). It is therefore reasonable to assume that for low frequencies the loop response will be dominated by the compensation filter. The RF components do however have a finite bandwidth. The finite bandwidth is caused by high frequency poles and zeros due to the filtering distributed across the RF components. The simplest way to reproduce both the low frequency requirements and high frequency characteristics is to model the loop compensation directly combined with a time delay. In the early stage of this work the delay was approximated from data sheets and RF component measurements. Later the delay was measured from the system implemented in section 3.4.

Consider the calculated bode response shown in figure 3.8 of a system with a single pole

p at a pole location frequency given in radians per second, a DC gain term K and a time delay τ given in seconds. The transfer function is given by

(3.2)

Assuming at this stage, the cartesian feedback components are wideband, linear and no cross-coupling exists between the I and Q paths (i.e δ= 0 in ejδ of figure 3.7), then the

G s( ) = _{s p}--- eKp₊ –τs

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pole represents the dominant pole purposely introduced by the baseband filters. The DC gain represents the loop gain which includes the gain of the baseband filters and the gain of the RF stages, and the delay concisely models phase shift introduced by high frequency poles and zeros in addition to actual transmissive delay. Using classical bode techniques it is therefore possible to determine the gain and phase margins for different combinations of gains and delay (table 3.1).

Figure 3.8: Bode Response of G(s) with a single pole at 20kHz, a DC gain of 85.2 (38.6dB), and a 50ns delay. The phase margin as drawn is 60°, and the gain margin as drawn is 9.4dB.

102 103 104 105 106 107 108 −40 −20 0 20 40 Frequency (Hz) Gain (dB) 102 103 104 105 106 107 108 0 −90 −180 −270 −360 Frequency (Hz) Phase (degrees)

Table 3.1: Some example G(s) transfer functions Pole Frequency, (kHz) Phase Margin, (degrees) Gain Margin, (dB) DC Gain System Delay (ns) 20 20 2.2 195.2 50 20 30 3.5 167.6 50 20 40 5.1 140.0 50 20 50 7.0 112.5 50 20 60 9.4 85.2 50 20 70 12.7 58.3 50 Frequency Response

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3.2.1 Gain Maximization

Since the forward loop gain reduces distortion produced in the PA, it is desirable to maximize gain over as large a bandwidth as possible. This can be achieved by introducing more elaborate compensation transfer functions consisting of many poles and zeros. Figure 3.9 shows a possible alternative compensation filter response (solid lines) compared to the single pole compensation described previously (dot-dashed lines). With two poles and one zero, the loop gain (and hence distortion reduction) has been increased by 10dB whilst the stability as measured by the gain and phase margins is essentially unchanged. The main drawback is that the phase response indicates relatively less stability over one decade.

Although increasing the compensation complexity has some benefits in terms of increasing loop gain, single pole compensation was favoured in this research in order to facilitate the comprehensive stability analysis given in the next chapter.

Figure 3.9: Bode response comparison of loop compensation filters G(s). The dot-dashed lines give the bode response of the single pole (20kHz) and delay (50ns) previously described. The solid lines give the bode response of a compensation filter with two poles at 20kHz, a zero at 65kHz, a 50ns delay and 10dB more gain than the single pole filter. The phase margin as drawn is 59°, and the gain margin as drawn is 9 6dB 102 103 104 105 106 107 108 −40 −20 0 20 40 Frequency (Hz) Gain (dB) 102 103 104 105 106 107 108 0 −90 −180 −270 −360 Frequency (Hz) Phase (degrees) Gain Maximization

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