Multirate Signal Processing

In single-rate systems, only one sampling rate is used throughout a digital signal processing system, whereas in multirate systems the sampling rate is changed at least once. This laboratory deals with the realization and analysis of multirate systems. In the first experiment we will examine the effects of decimation and interpolation. Furthermore, a small multirate signal processing system is built and the quality of the reconstructed output signal will be discussed. 1 Decimation and Interpolation Decimation and interpolation are processes that transform a discrete-time signal with sampling rate fs to another discrete-time signal with a new sampling rate f ′ s. A decimator realizes a sampling rate conversion (down-sampling) by an integer factor M : f ′ s = fs/M , by using only every M th input signal sample in the output signal. The output signal is at a lower sampling rate, and thus has a lower bandwidth (f ′ s/2) than the input signal (fs/2). To avoid aliasing and thus ensure correct reproduction of the signal spectrum in f ′ s/2, the input signal has to be low-pass filtered before sampling rate conversion. The block diagram of a decimator and example spectra of input, intermediate, and output signal for a decimation factor M = 3 are shown in Figure 1. Note the different scaling of the normalized frequency θ′ axis in the output signal spectrum. For interpolation a sampling rate conversion (up-sampling) by an integer factor L is achieved by inserting L − 1 zero samples between adjacent input signal samples. The resulting signal has a sampling rate f ′ s = Lfs, and a spectrum that includes periodic images of the original signals spectrum. To correctly reproduce the spectrum of the input signal (in the frequency range up to fs/2) subsequent low-pass filtering is necessary to avoid imaging [1]. The block diagram of an interpolator, example signals and spectra for an interpolation factor L = 3 are shown in Figure 2. Fractional changes L M of the sampling rate can be achieved by combining a decimator with factor M with an interpolator with factor L. For block processing the realization of decimation/interpolation is trivial, as long as the block size is an integer multiple of the decimation/interpolation factor. Otherwise, further considerations are necessary (scheduling, block redistribution, delay, etc.). Multirate Signal Processing 2