A Fundamental Limitation of DC-Free Quantization Noise With Respect To Nonlinearity-Induced Spurious Tones

Abstract

Fractional-N phase-locked loops (PLLs) are widely used to synthesize local oscillator signals for modulation and demodulation in communication systems. Such PLLs generate and subsequently lowpass filter DC-free quantization noise as part of their normal operation. Unfortunately, the quantization noise and its running sum inevitably are subjected to nonlinear distortion from analog circuit imperfections which causes spurious tones in the PLL output signal that can degrade communication system performance. This paper presents the first general mathematical analysis of this phenomenon. It proves that if the running sum of the quantization noise, t[n], satisfies t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">low</sub> <; t[n] ≤ t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">high</sub> for all n, where t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">low</sub> and t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">high</sub> are integers, then subjecting t[n] to kth-order distortion for at least one k ∈ {1, 2, 3..., t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">high</sub> - t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">low</sub> } will result in spurious tones for most fractional-N PLL output frequencies regardless of how the quantization is performed. It also shows that quantizers exist which are optimal in the sense that subjecting the running sum of their quantization noise to th-order distortion for any k ∈ {1, 2, 3..., t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">high</sub> - t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">low</sub> - 1} does not result in any spurious tones. In a typical fractional-N PLL, the larger the range of t[n] the greater the power of the PLL's phase noise, so these results imply a fundamental tradeoff between phase noise power and spurious tones in PLLs.