The readout signal of a body dithered ring laser gyro contains both useful information and the dither component. The dither component must be removed to get the useful information. The dither stripping method can get the useful information without latency. But the quadrature demodulator only has $1/4$ pulse resolution, which means that the quantization noise will limit the short-term accuracy. The effect and the property of quantization noise are analyzed in detail. Taking advantage of high-speed analog-to-digital conversion (ADC), the analog values of the two beat frequency signals in quadrature are sampled. A novel instantaneous phase method is introduced, which can get the residual phase besides $4\times$ resolution and the quantization noise can be removed radically. The Allan variance analysis of experimental results shows that the quantization noise coefficient using this method is only $1/7$ of that using the original $4\times$ resolution method.