The present disclosure relates to signal processing filters.
A signal processing filter is a circuit that can be used to remove or attenuate undesired frequency components from an analog signal, to enhance desired frequency components, or both. A filter's frequency response, e.g., high-pass, low-pass, bandpass, notch, or all-pass, is defined primarily by its transfer function. The transfer function H(s) of a filter is the ratio of the output signal VOUT to the input signal VIN as a function of the complex frequency s as given by the equation
      H    ⁡          (      s      )        =                    V        OUT            ⁡              (        s        )                            V        IN            ⁡              (        s        )            where s=σ+jω. The number of poles in the transfer function can determine the order of the filter. A high order filter will have a frequency response with a steeper slope than a low order filter. To achieve the desired attenuation of undesired frequency components that are close to a desired frequency component, a high order filter may be needed.
A multiple feedback filter, also known as a Rauch filter, can be used to implement a second order low-pass filter. FIG. 1 shows a conventional Rauch filter 100. The Rauch filter 100 receives a single input signal VIN and provides a single output signal VOUT using an operational amplifier or operational transconductance amplifier 102. The transfer function of the Rauch filter 100 is given by the following equation:
      H    ⁡          (      s      )        =                    (                              G            2                    ⁢                      G            3                          )            /              (                              C            1                    ⁢                      C            2                          )                            s        2            +                        s          ⁡                      (                                          G                1                            +                              G                2                            +                              G                3                                      )                          /                  C          1                    +                        (                                    G              1                        ⁢                          G              2                                )                /                  (                                    C              1                        ⁢                          C              2                                )                    where G1=1/R1, G2=1/R2, and G3=1/R3. The transfer function of the Rauch filter 100 has two poles and real coefficients in the transfer function. The Rauch filter 100 can be included in, for example, a global positioning system (GPS) receiver for processing GPS signals.
FIG. 2 shows a fully differential Rauch filter 200 that receives a differential input signal including signals IP and IN and provides a differential output signal including signals OP and ON using a fully differential operational amplifier or operational transconductance amplifier 202. The differential output signal of the Rauch filter 200 is controlled by two feedback paths. The transfer function of the Rauch filter 200 has two poles and real coefficients in the transfer function.
In applications such as low radio frequency (RF) communication receivers with low-IF (intermediate frequency) downconversion, a complex bandpass filter with an asymmetric frequency response with respect to frequency f=0 may be used to separate the desired radio frequency signal from all other signals picked up by an antenna. A complex filter can be used to implement a bandpass filter that has an asymmetric frequency response. A complex filter has a transfer function with complex coefficients which correspond to the asymmetric frequency response.
An example of a complex bandpass filter includes an active RC filter 300 as shown in FIG. 3. The active RC filter 300 includes a pair of amplifiers, for example, operational amplifiers or operational transconductance amplifiers 302 and 304. The amplifier 302 processes a complex signal that includes pairs of real signals. The first pair of real signals, which is the real component of the complex signal, includes signals received at inputs IP and IN. The second pair of real signals, which is the imaginary component of the complex signal, includes signals received at inputs QP and QN. The cross-coupling of the real and imaginary signal paths using resistors 306, 308, 310, and 312 results in the transfer function having complex coefficients.
The active RC filter 300 is a first order complex bandpass filter that has a transfer function with one pole. If an undesired signal has a frequency near the frequency of the desired signal, a first order bandpass filter may not provide sufficient attenuation of the undesired signal. A second order complex bandpass filter can be implemented using two pairs of amplifiers by, for example, cascading two stages of the active RC filter 300. Because such a filter includes four amplifiers, the filter may increase the size and power consumed by a system, such as a low-IF RF signal receiver, that includes the filter.