Multirate Digital Signal Processing
Contents 1) What is multirate DSP? 2) Downsampling and Decimation 3) Upsampling and Interpolation 4) FIR filters 5) IIR filters a) Direct form filter b) Cascaded form filter 6) Polyphase filters 7) Advantages of multirate DSP 8) Applications of multirate DSP a) Design of phase shifts b) Interfacing of digital systems with different sampling rates c) Implementation of digital filter banks d) Subband coding of speech signals e) Quadrature mirror filters (QMFs) f) Transmultiplexers g) Oversampling A/D and D/A conversion
Introduction Interest in signal processing long predates computers. As long as people have tried to send or receive information through electronic media, such as telegraphs, telephones, television, radar, etc., there has been the realization that these signals may be affected by the system used to acquire, transmit, or process them. Sometimes these systems are imperfect and introduce noise, distortion, or other artifacts. Understanding the effects these systems have and finding ways to correct them is the foundation of signal processing. There are many types of signal processing. Among those Digital signal processing is more efficient and widely used. Multirate systems are building blocks commonly used in digital signal processing (DSP). Their function is to alter the rate of the discrete-time signals, which is achieved by adding or deleting a portion of the signal samples. "Multirate" simply means "multiple sampling rates". A multirate DSP system simply uses more than one sampling rate within the system. In many systems, multrate DSP increases processing efficiency, which reduces DSP hardware requirements. Also, a few systems are inherently multirate, for example, a "sampling rate converter" system that converts an input sampling rate to a different output sampling rate. Multirate systems play a central role in many areas of signal processing, such as filter bank theory and multiresolution theory, they are essential in various standard signal-processing techniques such as signal analysis, denoising, compression and so on. During the last decade, however, they have increasingly found applications in new and emerging areas of signal processing, as well as in digital communications.
"Multirate" means "multiple sampling rates". A multirate DSP system uses multiple sampling rates within the system. Whenever a signal at one rate has to be used by a system that expects a different rate, the rate has to be increased or decreased, and some processing is required to do so. Therefore "Multirate DSP" refers to the art or science of changing sampling rates. "Resampling" means combining interpolation and decimation to change the sampling rate by a rational factor. Resampling is done to interface two systems with different sampling rates. Ex: Professional audio equipment uses a sampling rate of 48 khz, but consumer audio equipment uses a rate of 44.1 khz. To transfer music from a professional recording tape to a CD, the sampling rate must be changed by a factor of 44100 / 48000 = 441/480=147/160.Therefore we would interpolate by a factor of L=147 then decimate by a factor of M=160.The resampling factor is 147 / 160 = 0.91875.The Nyquist criteria must be met relative to the resulting output sampling rate to prevent aliasing. Since resampling includes interpolation and decimation, we require an interpolation and a decimation filter Multirate DSP consists of: 1. Decimation: It is a process to decrease the sampling rate. 2. Interpolation: It is a process to increase the sampling rate. "Downsampling" is a process of removing some samples, without the lowpass filtering. A signal is downsampled only when it is "oversampled"(i.e. sampling rate > Nyquist rate). This combined operation of filtering and downsampling is called Decimation. To downsample by a factor of M, we must keep every Mth
sample as it is and remove the (M-1) samples in between. Ex: To decimate by 4, keep every fourth sample, and remove three out of every four samples. Symbol of downsampler The graphical representation for M=4 is Block diagram of a decimator "Upsampling" is the process of inserting zero-valued samples between original samples to increase the sampling rate. (This is called "zero-stuffing"). Given a sequence x[n], we can define Where xu[n] is the sequence up-sampled from x[n] by a factor of L.This means that xu[n] is generated by padding (L-1) zeros between every sample of x[n]. Symbol for up-sampler Graphical representation for L=2
"Interpolation" is the process of upsampling followed by filtering (to remove the undesired spectral images.) The result is a signal sampled at a higher rate. The interpolation factor (L) is the ratio of the output rate to the input rate. Block diagram of an interpolator Interpolation consists of two processes: 1) Zero stuffing :Inserting (L-1) zero-valued samples between each pair of input samples. The zero stuffing creates a higher-rate signal whose spectrum is the same as the original over the original bandwidth, but has images of the original spectrum centered on multiples of the original sampling rate. 2) Lowpass-filtering: The lowpass filtering eliminates the images. There are 3 different kinds of filters. They are: 1) FIR filter 2) IIR filter : a) Direct form b) Cascaded form 3) Polyphase filter
1) FIR filter: A causal FIR filter has the following difference equation Where M is the order. The result y[n] is the discrete convolution of x[n] with the (finite) impulse response: 2) IIR filter: The input x[n] and output y[n] of a causal IIR filter satisfy the Nth order linear constant-coefficients difference equation of the form. Often the coefficient a0 is assumed to be 1 and we can rewrite the difference equation as Where k=1,2 N. a) Direct form: The system function is of the following form.
b) Cascaded form: The system function is One advantage of cascaded form over the direct form is that a small change of a coefficient (ex:quantization)moves only the pair of poles(or zeros)of the corresponding stage and not all others.furthermore,the amount of displacement is less than for the overall higher order direct form filter. Polyphase filtering Polyphase filtering is a technique that allows us to reduce the computational requirements when performing convolution followed by down sampling. For example, consider the basic lowpass filtering followed by down-sampling structure: This direct implementation is extremely inefficient since the tapped delay line is computing all the samples at its output and yet (M-1) of them are thrown away. Advantages of Multirate DSP. 1) With interpolation and decimation, the computational and/or memory requirements of the resampling filtering can sometimes be greatly reduced by using multiple stages.
2) Sampling rate conversion of a digital signal can be accomplished in two methods. One method is to pass the digital signal through a D/A converter, filter it and then resample the resulting analog signal at the desired rate. The second method is to perform the sampling rate conversion entirely in the digital domain. The advantage of the first method is that the new sampling rate can be arbitrarily selected and needn t have any special relationship to the old sampling rate. 3) Multirate Digital signal processing is more efficient, distortion less and flexible type of signal processing. Applications of Multirate digital signal processing 1) Used for the design of phase shifters 2) Interfacing of digital systems with different sampling rates 3) Implementation of digital filter banks Filter banks are used for performing spectrum analysis and signal synthesis. The filter banks are basically two types. They are Analysis filter banks and Synthesis filter banks. a) Synthesis filter bank b) Analysis filter bank
4) Subband coding of speech signals. Multirate signal processing notions provide efficient implementations of the subband encoder. Subband coding is a method, where the speech signal is subdivided into several frequency bands and each band is digitally encoded separately. Subband coding is also an effective method to achieve data compression in image signal processing. 5) Quadrature mirror filters (QMF). Two channel QMF bank. QMfs split the input signal into two output signals with bandwidth half of the original bandwidth. Thus the sampling rate of the output signals can be decimated by a factor of two. The output of the QMF filter bank after being processed (encoding, decoding, individual amplification etc) is recombined to a single signal using the synthesis filter bank also composed of QMFs. 6) Transmultiplexers. These are the devices used for converting between Time Division Multiplexed (TDM) signals and Frequency Division Multiplexed (FDM) signals. 7) Oversampling A/D and D/A conversion. An oversampling A/D converter is implemented by a cascade of an analog sigma-delta modulator (SDM) followed by a digital anti-aliasing decimation filter and a digital
highpass filter. The analog SDM produces a one-bit per sample output at a very highsampling rate, which is passed through a digital lowpass filter, which provides a high precision output that is decimated to a lower sampling rate. This output is then passed to a digital highpass filter that serves to attenuate the quantiztion noise at the lower frequencies. The digital signal is passed through a highpass filter whose output is fed to a digital interpolator. This high sampling rate signal is the input to the digital SDM that provides a high sampling rate, one-bit per sample output, which is then converted to an analog signal by lowpass filtering and further smoothing with analog filters.