GmC integrators for lowpower and low voltage applications. A gaussian polyphase filter for mobile transceivers in 0.35
4.5. Lowpower, gaussian, polyphase filter for mobile transceivers. Matching driven power.
We have discussed so far applications where noise driven power consumption is dominant. There are filter applications where matching requirements and noise requirements have the same importance, with constraints on power consumption and linearity. Channel selectivity in receivers has been realized until recently, using SAW filters. Those components are external components and therefore integration on chip of selectivity has become a major concern in receivers. From Chapter 3 we already know that selectivity increases the noise power and requires extra power consumption to achieve it. A polyphase filter is an example of a selective filter without the need of using high Q bandpass sections. Here selectivity is rather ensured by using polyphase signals and two lowpass filter sections where matching driven power consumption comes as a variable. Polyphase filters can discriminate between positive and negative frequencies and therefore, using this property, selectivity can be achieved [15]. By using a low power integrator, we are going to show how to realize a polyphase filter needed for image rejection in a mobile transceiver.
4.5.1. The low IF receiver topology and polyphase filters
As mentioned before, we want to discriminate between positive and negative frequencies in order to realize on chip selectivity. This is not possible with real signals but with two dimensional signals or complex signals. We can imagine positive and negative frequencies as being phasors rotating in the complex plan in opposite direction [16]. The complex signals used in a receiver are called polyphase signals which consist of a number of real signals with different phases. A quadrature signal consists of two real signals with p/2 phase shift:
Fig.4.17: Low IF receiver topology 
(4.25)
Based on complex notation it is possible to define operations like multiplication, and convolution for complex signals with multipliers, filters and amplifiers for complex signals. For more details on polyphase signals references [17], [18], [19] and [20] can be consulted.
In order to show the connection of complex signals with polyphase filters and low IF receivers, consider fig.4.17 where the low IF concept is shown. A broadband HF filter is used to prevent overloading of the mixers with strong out of band signals. The low noise amplifier (LNA), amplifies the weak signal from antenna to come to a sufficient signaltonoise ratio. The mixers are downconverting the signal to a low intermediate frequency IF by multiplication with two quadrature signals which can be seen as a single positive frequency e^{jwt}.
The polyphase bandpass filter ensures the rejection of the mirror frequency and provides the antialiasing necessary in the digital signal processor (DSP) which does the final downconversion to baseband and demodulation of the signal. The downconversion spectra are shown in fig.4.18. The wanted signal is multiplied with a single positive frequency at f_{LO}. The mirror signal will be mixed down from f_{mirror} to f_{IF} and the wanted signal at f_{IF}.
With a polyphase filter it is possible to discriminate between the negative and positive frequencies and therefore, the mirror frequency will be filtered out. The advantage of this topology consists in the high level of integration and the lack of DC offsets as in the case of zero IF receiver. The second IF is at low frequency and digital signal processing is possible. Different receivers have different specifications and that is why we want to find the specifications for the polyphase filter from the receiver specifications.
Fig.4.18: Low IF downconversion 
4.5.2. Specifications
The most common standards in use nowadays are DECT, GSM and DCS1800. The specifications in terms of frequency range, selectivity and channel spacing are given in Table 4.5. For further details reference [UCB] can be used.
DECT 
GSM 
DCS 1800 

Frequency range [MHz] 
18001900 
925990 
18051880 
Channel spacing [MHz] 
1.7 
0.2 
0.2 
Noise figure 
15dB 
10.3dB 
12.3dB 
IP3 
25dBm 
18dBm 
13dBm 
Selectivity n=1 
16dB 
18dB 
18dB 
Selectivity n=2,3,... 
36dB 
43dB 
43dB 
Table 4.5: Receiver specifications for different standards
The focus of this section is the design of a polyphase filter for a lowIF DECT receiver. In this standard, the channel spacing is 1.7MHz and the signal bandwidth is about 800KHz. The central frequency is chosen at 1MHz whereas the rejection of the filter at 2.3MHz should be at least 16dB.The image rejection should be 16dB and the image is located 2MHz away from the central frequency. In order to achieve the image rejection spec’s also for GSM standards we are requiring an image rejection of better than 40dB.
4.5.3. Filter synthesis
In data communications, the group delay specifications and the step response of the filter are very important. The gaussian transfer offers a flat group delay characteristic and virtually no overshoot in the step response. Gaussian filters are presented in Appendix 2. When gaussian approximation is used, a seventh order filter is necessary. That is why the gaussianto6dB approximation will be used and
Fig.4.19: Polyphase lowpass to bandpass transformation 
therefore a fifth order filter will suffice. The price paid is a deterioration of the group delay and little overshoot which can be tolerated. The transfer function to be synthesized [21] is:
(4.26)
The low pass prototype is shown in fig.4.19. The transformation from a lowpass into a polyphase bandpass characteristic is achieved by using complex network elements as illustrated in the same picture. Every capacitor is replaced with a parallel connection of a capacitor and a frequency dependent negative element. A coil will be replaced by a parallel connection of a coil and a frequency dependent negative element. The result is shown in fig.4.20 with chosen state variables.
The signal flow graph SGF or the leapfrog simulation of ladders method is a direct way to convert the prototype into a filter realization. The result is a filter with low sensitivity to mismatch and realizable with transconductors and grounded capacitors. The state variables I_{kx} and I_{k} (k=1,3,5) represent the separation between a complex network element and a real network element. The imaginary part or the complex network element is generated by copying currents from the quadrature signal path with a copying factor dependent on the value of the element and the center frequency of the bandpass filter. This is shown explicitly in fig.4.20 where the source and the load transformation for a current input has been already performed. The filter consists actually from two coupled lowpass sections for the I signal and for the Q signal respectively.
Fig.4.20: The SFG of the polyphase filter 
It is to be mentioned the quadrature inputs/ outputs and the quadrature state variables I_{k} and jI_{k} (k=1,3,5) in the signal flow graph. The block diagram of the complete filter realized with current GmC integrators is illustrated in fig.4.21.
The reason for choosing current GmC integrators, for realization, is their power efficiency. Other reasons will be discussed later on. The copying factors A_{1...5} are realized as aspect ratios in the basic integrator from fig.4.9 where the core transconductance element is the same for all integrators the only difference being only the capacitance values and different coupling factors A_{k} for every section. The advantage of using equal transconductances consists in better matching between integrators.
Fig.4.21: Block diagram of the polyphase filter 
4.5.4. The effect of mismatch on polyphase signals
Mismatch in amplitude and/or phase will impair the polyphase signals which consist of two real signals with a constant phase difference. The mismatch in amplitude and phase generates frequency crosstalk between negative and positive frequencies. In fig.4.22, the RF signal which consists of two quadrature signals is mixed with the local oscillator quadrature signals. The phase mismatch can be added to the two oscillator signals whereas the amplitude mismatch is added in the polyphase filter. The gain mismatch and the amplitude mismatch generate two signals S_{I}(t) and S_{Q}(t) at intermediate frequency f_{IF}, affected by errors:
(4.27)
Since the errors are small, we can approximate the signals as shown in fig.4.22. In the complex plane, the ideal signals are represented as empty circles and the black circles are signals with errors. Magnitude errors will generate crosstalk between the positive and negative frequency components while phase errors in the quadrature outputs of the oscillator generates crosstalk between the I and Q signals. For simplicity, consider only the gain mismatch and I=Q=1. The complex signal at the output can be found from:
(4.28)
Fig.4.22: Gain and phase mismatch in polyphase signals 
Besides the wanted positive component, there is a negative frequency component proportional to the amplitude mismatch. A mismatch of 2% gives a crosstalk of about 40dB. Another cause of error is the phase mismatch in the polyphase bandpass section. Given the quadrature signals I and Q impaired only by phase errors:
(4.29)
then, the complex signal at the output with errors will be:
(4.30)
For a phase difference D f /2 of 1° we get a frequency crosstalk of 35dB. This explains why the requirements on phase and amplitude mismatch in the polyphase filter are very strong in order to achieve the required spec’s.
To be mentioned the need to match only the two real integrators in the polyphase integrator. If this condition is fulfilled, the quadrature nature of the two outputs is preserved. A limited quality factor of the integrators gives amplitude errors as explained in eq.(4.13). By using an OTA the nondominant poles and zeros degrade the quality factor giving amplitude errors. The most difficult constraint will be to match two OTA’s while the number of components is large (common mode circuit plus differential stage). The power consumption, area and the supply voltage are also limiting factors in using OTA’s or Opamps to realize the filter.
4.5.5. Filter realization and simulation results
The filter has been realized with integrators from section 4.3. The central frequency of the filter can be easily tuned by changing all the bias currents of the integrators simultaneously. The simulated polyphase transfer for different bias currents I_{0}=I_{BIAS} is shown in fig.4.23.
The dimensions of the transistors and the scaling factors of every integrator are given in table4.6. The input transistors are scaled in the same way as the polyphase coupling stages with a factor A_{k} (k=1..5). The I and the Q outputs of the integrator have the dimensions given in the first two columns from table 4.6. In order to study the negative frequency crosstalk, Monte Carlo simulations have been performed.
(W/L)_{nMOS} 
(W/L)_{pMOS} 
A_{1} 
A_{2} 
A_{3} 
A_{4} 
A_{5} 

8/4 
40/10 
1.1 
2.1 
2.7 
2.6 
6 
Table 4.6: The dimensions of the polyphase sections
Fig.4.23: Tunable polyphase transfer 
For simulations, we have considered matching between the transistors connected at the same gate potential and also matching between the transistors occupying the same position in the differential polyphase integrator. In this approach, due to the simplicity of the integrator and the small dimensions used for the transistors it is very easy to ensure matching as explained above. This is another reason to chose for the integrators from section 4.3. The power supply voltage was set to V_{DD}=2.5V and the bias current is I_{0}=10m A. The results of the simulation are shown in fig. 4.24.
The image frequency will be at 1MHz and the image rejection here is about 40dB. Due to the large quality factor of the integrators, the magnitude errors are below 0.5dB in the passband. The power consumption of the filter, in this case is 5mW.
Fig.4.24: Negative frequency crosstalk 
Fig.4.25: Polyphase differential integrator with source degeneration 
The image rejection can be increased further by using source degeneration polysilicon resistors as shown in fig.4.25. Matching between transistors can be improved at the expense of voltage swing and the area penalty for using resistors. Another effect will be the reduction of the transconductance of the transistors with a factor (1+g_{m}R) given by series feedback loop gain. The V_{T} mismatch between transistors is desensitized with the same factor. The minimum supply voltage can be as low as V_{DDmin}=1.5V with
Fig.4.26: Negative frequency crosstalk (source degeneration) 
a nominal supply voltage of 2.5V. The results of the simulation are illustrated in fig. 4.26. The power consumption in this case is 15mW for the filter. The image rejection is 52dB at the expense of higher power consumption and less swing. The value of the resistors is about 25KW scaled according to the current levels. The value of the integration capacitors varies between 3.9pF for the first section and 20pF for the last section scaled according to the A_{K} factors (k=1…5). The passband amplitude ripple is increased to 1dB. This can be explained from the decrease in the output resistance of the current sources at higher current levels and the decrease of the integrator gain. The result is the decrease in the quality factor of the integrator.
In references [22], [23] a polyphase filter based on opamps has been presented. The filter is tuned from switched capacitor banks. Here matching is achieved using polysilicon resistors and due to the use of opamps the dynamic range is high. Although the mirror signal suppression is better than 60dB, the power consumption is 90mW and the large dynamic range is obtained by using automatic gain control. To be mentioned the power supply voltage of V_{DD}=5V. The total area is 7.5mm^{2} showing the main disadvantage of that approach, namely power consumption and area.
4.5.6. Noise properties of the filter
The filter has been terminated with a load resistor and the total voltage noise power on the load resistor has been simulated. The noise has been integrated in the bandwidth of the filter 500KHz…1.5MHz as illustrated in fig.4.27. The scaling factors represent the W/L scaling factors from Chapter 3. The aspect ratios of the transistors and capacitors are going up s times, the resistors are going down s times and the current level increases s times without changing the voltage levels in the filter. The dynamic range for s=1 is 69dB. A factor two increase in scaling factor s gives a factor two improvement in the noise power and 3dB improvement in the dynamic
Fig.4.27: The noise power of the filter for different scaling factors 
range as expected. The matching will be also improved by scaling the filter. In conclusion this approach is attractive for low voltage supply in terms of power, area, complexity and time domain properties when comparing to opamp approach.
