Method and apparatus for complex cascade sigma-delta modulation and single-sideband analog-to-digital conversion

A complex cascade sigma-delta modulator for analog-to-digital conversion applications. The modulator includes first and second sigma-delta modulator stages, combined with a complex digital noise cancellation circuit. In addition, analog-to-digital conversion of baseband signals using a complex sigma-delta modulator is also presented.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to complex analog-to-digital converters, and more specifically to methods and apparatus for complex sigma-delta modulation.

2. Discussion of the Related Art

Analog-to-digital converters (ADCs) are used to convert analog information to digital information so that signal processing may be accomplished in the digital domain. In particular, sigma-delta ADCs are useful in such applications. Sigma-delta ADCs convert incoming analog signals in a particular frequency span of interest into a high-rate (oversampled), low resolution (one-bit) digital output data stream. The sigma-delta approach to analog-to-digital conversion is well-known for its superior linearity and anti-aliasing performance compared to traditional ADC conversion approaches with lower sampling rates.

In order to maintain the full performance of sigma-delta conversion, it is desirable to implement a “complex” converter, which may be thought of as converting a pair of input signals into streams of digital output values, one such stream representing the “real” or “in-phase” (I) component of the signal, and the other such stream representing the “imaginary” or “quadrature” (Q) component of the signal. It is convenient and common to represent the two output data streams I and Q as a single complex data stream I+jQ, where j is a symbol representing the square root of −1.

The advantages of sigma-delta modulators come at some expense. For example, the quantization of the signal produces noise in the output data stream, known as quantization noise. An important job of a sigma-delta converter is to “shape” this quantization noise out of the frequency range which contains the desired signal, so that subsequent digital filtering operations may recover the desired signal without corruption. In a subsequent stage, this out-of-band quantization noise may be eliminated by means of a filter. In the case of a low-pass sigma-delta modulator, the band of interest spans a frequency range centered around DC, as shown inFIG. 1A, whereas in a bandpass sigma-delta modulator, the center frequency is shifted to a higher frequency, as shown inFIG. 1B.

Two basic possibilities to improve the performance of sigma-delta modulators are the use of a higher-order modulator, or the use of a multi-bit quantizer. These approaches are not necessarily the most effective solutions. The former leads to system instability and latter may cause non-linearity. Cascading of low-order single-bit modulators has been proven to be an efficient way to achieve a higher performance without facing the above-mentioned problems. Cascaded modulators require a digital noise cancellation circuit to remove the quantization noise introduced by the first stages. Consequently, the output quantization noise will be ideally due to the very last stage of the modulator.

As may be seen in referencesFIGS. 1A and 1B, quantization noise101,102+ and102− and the desired signal105,106+ and106− all remain symmetric with respect to the vertical axes103,104for both the low-pass and bandpass modulators. This way of shaping the quantization noise is wasteful when one side of the spectrum, for example, positive frequencies, provides all the information carried by the signal. For instance, the quanrature bandpass signal shown inFIG. 2exhibits such a property. This property has been the main motivation for using complex sigma-delta modulators for quanrature bandpass signals. A complex sigma-delta modulator may be implemented using two real modulators with the interconnections between them in such a way that the output complex signal, yr+jyi, exhibits an asymmetric spectrum for quantization noise.

However, it is not possible to use the same principle for baseband signals because the complex signal I+jQ has spectral content at both positive and negative frequencies. For this reason, only real sigma-delta modulators with a symmetric noise shaping characteristic have been used for direct conversion systems, and two real sigma-delta modulators have been required to process the in-phase and quadrature components.

A single complex modulator is far more efficient in terms of noise shaping than two real modulators operating separately with xr=I and xI=Q. In other words, for a given number of integrators, a complex sigma-delta modulator provides a better signal-to-noise (SNR) ratio. Alternatively, for a given SNR, a complex modulator requires a smaller number of integrators. This, in turn, translates into a smaller chip area and lower power consumption. The main issue with both cascade and complex modulators is their sensitivity to variation of coefficients. Inaccuracy of the coefficients in a complex modulator degrades the quality of noise shaping and causes image leakage. In a cascade structure, mismatch between the coefficients of the modulator and the coefficients of the digital noise cancellation circuit limits the achievable SNR.

DETAILED DESCRIPTION

The invention and the various features and advantageous details thereof are explained more fully with reference to the nonlimiting embodiments that are illustrated in the accompanying drawings and detailed in the following description. It should be understood that the detailed description and the specific examples, while indicating specific embodiments of the invention, are given by way of illustration only and not by way of limitation. Various substitutions, modifications, additions and/or rearrangements within the spirit and/or scope of the underlying inventive concept will become apparent to those of ordinary skill in the art from this disclosure.

The present invention solves the above-noted discrepancies in prior approached to sigma-delta modulation. One aspect of the invention contemplates a complex cascade sigma-delta modulator comprised of at least two cascaded complex sigma-delta modulators combined with a digital complex noise cancellation circuit. In addition, another aspect of the invention is the use of complex sigma-delta modulators for baseband signals having negligible frequency-spectral content about DC. This results in higher efficiency in analog-to-digital conversion because the required bandwidth of the sigma-delta modulator is halved. In addition, it provides a sigma-delta modulator that enjoys high performance without at the same time suffering inaccuracy due to variation of coefficients. Furthermore, it permits use complex sigma-delta modulators for baseband signals which results in simplified structure, cost and efficiency.

A complex signal x(t) may be represented by its real and imaginary parts as: x(t)=xr(t)+jxi(t). Similarly, the impulse response of a complex linear system, h(t), may be represented by its real and imaginary parts: h(t)=hr(t)+jhi(t). In the frequency domain, the transfer function of the system becomes H(z)=Hr(z)+jHi(z). Therefore, the complex system shown inFIG. 3Amay be implemented using two real systems. As an example, the complex integrator301shown inFIG. 3Bmay be realized as shown inFIG. 3C, by interconnecting two real integrators302,303. The complex output signal is similarly given by y(t)=yr(t)+jyi(t). The complex integrator301has a complex pole at p=1+d+jc, where d and c are selectable coefficients.

Based on this concept of complex systems, the second-order complex sigma-delta modulator shown schematically inFIG. 4Amay be implemented using four real integrators as shown inFIG. 4B.

Referring toFIG. 4A, complex sigma-delta modulator400is comprised of two complex integrators401,402, combined with a complex quantizer403. Referring toFIG. 4B, in a real implementation, and using the principals discussed above with reference toFIGS. 3B and 3C, complex integrator401may be implemented using two real integrators,404,405, complex integrator402may be implemented using two real integrators406,407, and complex quantizer403may be implemented using two real quantizers408,409.

A real fourth-order cascade sigma-delta modulator500implemented using two second-order modulators is shown inFIG. 5. This system uses four real integrators,501–504, to implement a 2-2 combination. Other combinations of real integrators, such as 1-1-1 and 2-1 to build a third-order system, or a 2-1-1 combination to build a fourth-order system, are also possible. The extension to even higher order systems is also possible. The system also employs real quantizers506,507, and real digital to analog converters.508,509. The outputs of the first and second stages are combined through a real noise cancellation circuit505to produce a final output Y(z).

In accordance with one aspect of the present invention, asymmetric noise shaping and cascading are combined in order to achieve a higher performance. The complex structure creates a single notch for quantization noise and cascading makes this notch deeper and wider. The general structure of a 2-2 cascade sigma-delta modulator600embodying one aspect of the present invention is shown in block diagram form inFIG. 6. The complex modulator includes a first-stage601cascaded with a second-stage602. The first-stage601includes complex integrators603and604, and quantizer606, and second-stage602includes complex integrators607,608and quantizer609. The modulator also includes a complex digital noise cancellation circuit611.

The output of the first-stage601, Y1(z) Y2(z), and the second-stage602, Y4(z), inFIG. 6are described below in equations 1 and 2.

The output complex signal is obtained by combining the complex output signals of each stage601,602using a complex noise cancellation circuit611with a complex transfer function NC(z):
Y(z)=z−2Y2(z)−NC(z)Y4(z)  (7)

Since p1and p2are both complex, NC(z) is also complex. The choice made in (3) and (4) allows the noise cancellation circuit611to be an FIR filter. The noise transfer function of the system is,
NTF(z)=(1−p1z−1)(1−p2z−1)(1−p3z−1)(1−p4z−1)  (10)

The four poles of the modulator, p1, p2, p3, and p4, may be all at a single frequency or may be distributed in an optimum fashion so as to maximize its SNR.

As stated above, p1=1+d1jc1andp2=1+d2jc2. Thus, the noise cancellation transfer function may be expressed as: NC(z),=NCr(z) +jNCi(z) where,

Therefore, the coefficients c1and c2used for noise cancellation circuit 611 preferably match with the same coefficients used in the first stage 601 of the modulator 60. Any mismatch between may degrade the performance of the system. The real implementation of the complex cascade modulator in accordance with one exemplary embodiment of the invention is shown inFIG. 7.

Referring toFIG. 7, complex cascaded sigma-delta modulator600may be implemented using real modulators and quantizers. For example, the first stage601of modulator600may include real integrators701–704and real quantizers705,706. Similarly, the second stage602of modulator600may include real integrators707–710and real quantizers711,712.

It should be noted that the system shown inFIG. 7may be implemented in hardware, software and/or firmware, or a combination of hardware, software and firmware, without departing from the scope of the present invention.

Simulation results using Matlab-Simulink have shown that the complex cascaded sigma-delta modulator with complex noise cancellation system, constructed in accordance withFIG. 7exhibits an excellent noise transfer function (corresponding to an excellent SNR, while at the same time exhibiting excellent immunity to coefficient variation).

Referring toFIGS. 8A and 8B, presented are exemplary embodiments of practical applications of the invention in super-heterodyne receivers. In bothFIG. 8A, antenna801is coupled to low noise amplifier802which produces an amplified radio frequency signal. The output of amplifier802is applied to mixers803and804along with quadrature local oscillator signals, SIN ω1(t) and COS ω1(t). The outputs of mixers803and804together represent a complex input signal A+jB that is asymmetric relative to DC with the desired signal located at positive frequencies and the image of the desired signal located at negative frequencies (or vice versa).

InFIG. 8A, the complex input signal A+jB, is applied to a polyphase filter806having an asymmetric frequency response with high attenuation at the image frequency. The outputs of polyphase filter806are applied to complex cascade ADC807, which may have the structure and function of complex cascade sigma-delta ADC600described above. The digital outputs of complex cascade ADC807are then down-converted to baseband by complex mixer808, to which are also applied oscillator signals SIN ω2(t) and COS ω2(t). The outputs of complex mixer808, are the demodulated I and Q signals of the received signal, and may then be further processed, for example by DSP809.

Alternatively, inFIG. 8B, the outputs of mixers803and804, A+jB, are applied to real band pass filters810and811, and then to complex cascade ADC812. Once again, complex cascade ADC812may have the structure and function of complex cascade sigma-delta ADC600described in detail above. The digital outputs of ADC812are then down-converted to baseband by complex mixer813, to which are also applied oscillator signals SIN ω2(t) and COS ω2(t). The outputs of complex mixer813, are the demodulated I and Q signals of the received signal, and may then be further processed, for example by DSP809.

Referring now toFIG. 9, disclosed is another aspect of the present invention in the conversion of baseband signals that have no or insignificant low frequency components. In this application, the input signal x(t) (900) has only a real component, and the complex sigma-delta modulator901receives only the real signal, and the imaginary input is grounded. Alternatively, the real signal x(t) may be applied to the imaginary input of sigma-delta modulator901. In either case, the center frequency of the sigma-delta modulator901is required to be only half of the signal bandwidth. Complex modulator901may be designed so that quantization noise903is shaped to include only positive or only negative frequencies.

In one embodiment, sigma-delta modulator901may be a complex cascade sigma-delta ADC having a structure and function like that of complex cascade sigma-delta ADC600, described above. In another embodiment, sigma-delta modulator901may be a complex sigma-delta modulator, such as that shown inFIGS. 4A and 4B, or it may be a complex sigma-delta modulator of conventional design, such as those disclosed in U.S. Pat. Nos. 6,225,928 or 6,329,939, the disclosures of each of which are incorporated herein by reference.

The output905of the complex sigma-delta modulator901is a complex signal, having both real and imaginary components, which are filtered by a digital complex filter904which removes out-of-band quantization noise. Filter904is a complex filter because the desired frequency response902is not symmetrical about DC. Complex filter904is of conventional design. Therefore, the quantization noise,903, will be present only at positive frequencies in the signal bandwidth. By taking only the real output of filter904, the complete signal spectrum at both positive and negative frequencies may be obtained. Mathematically, the output of the complex filter904is a complex signal y(t)=yr(t)+jyi(t), where yr(t) and yi(t) are the real and imaginary parts of the output signal, respectively. Since the signal has no spectral components at negative frequencies, the imaginary part of the signal must be the Hilbert transform of the real part: y(t)=yr(t)+jH[yr(t)], where yr(t) is a real signal having a spectrum that is symmetric about DC.

It should be noted that complex filter904may be combined with other digital filters for improved efficiency. Decimation filtering may also be performed at various stages of digital filter904. In this application, a requirement of digital filter904is that it have high attenuation at negative frequencies in order to minimize the error introduced by unfiltered negative frequencies. Thus a sharp filter roll off at DC is preferred. For example, filter904may have a cutoff frequency at a positive frequency near DC. In this situation, frequency content between DC and the cutoff frequency may be lost, but this is of no concern if the signal being conditioned has no frequency components close to DC. For example, this embodiment of the invention has applicability to audio signals with no frequency components of interest below 50 Hz. and WCDMA signal with little spectral content below 10 kHz.

Referring now toFIG. 10, presented is an exemplary embodiment of a practical application of the invention in which baseband signals are ADC converted in a zero-IF (homodyne) receiver. InFIG. 10, antenna1001is coupled to the input of low noise amplifier1002which produces an amplified radio frequency signal that is applied to mixers1003and1004. Also applied to mixers1003and1004are local oscillator signals SIN ωLO(t) and COS ωLO(t). The outputs of mixers1003and1004are baseband signals that are applied to low pass filters1005and1006and then to the un-grounded inputs of complex sigma-delta ADCs1007and1008. Consistent with the operation of ADC901discussed above with reference toFIG. 9, ADCs1007and1008convert only half of the signal bandwidth. The digitized complex outputs of ADCs1007and1008are applied to complex digital filter1009which recovers the entire signal spectrum from the half-spectrum digital outputs of ADCs1007and1008. The outputs of complex digital filter1009are the I and Q signals components of the received signal, and may then be further processed, for example by DSP1010. It should be noted that complex digital filter1009may be a single structure as shown, or filter1009may be separate structures, with each structure filtering the complex output of one of the ADC's1007and1008. Still further, the functions of complex digital filter1009maybe incorporated into the functions of DSP1010.

The terms a or an, as used herein, are defined as one or more than one. The term plurality, as used herein, is defined as two or more than two. The term coupled, as used herein, is defined as connected, although not necessarily directly, and not necessarily mechanically.

All the disclosed embodiments of the invention disclosed herein can be made and used without undue experimentation in light of the disclosure. It will be manifest that various substitutions, modifications, additions and/or rearrangements of the features of the invention may be made without deviating from the spirit and/or scope of the underlying inventive concept. It is deemed that the spirit and/or scope of the underlying inventive concept as defined by the appended claims and their equivalents cover all such substitutions, modifications, additions and/or rearrangements.