S-72.333 Postgraduate Course in Radiocommunications Fall 2000 Sampling, interpolation and decimation issues Jari Koskelo 28.11.2000.
Introduction The topics of this presentation are sampling, interpolation and decimation from the simulation point of view. The presentation is focused on digital systems due to the fact that the world seems to be going the digital way. Analog parts are also present in digital systems, but from simulation point of view they are quite often considered as is, meaning that they are present, but they are thought to be just a bit pipe. When simulating communication systems the sampling rate must be chosen carefully in order to simulate the right things. The sampling theorem indicates that the sampling rate must be at least twice the highest frequency. Some consideration is also given to interpolation and decimation matters. These are often used to change the sampling rate between different components in a simulated system. Background Usually the communication systems to be simulated are carrier modulated bandpass systems. Let the carrier frequency be f c and the bandwidth B. Then the signal spectrum is in the band f c B/2 f f c +B/2. A simple application of the sampling theorem indicates that sampling rate must be at least twice the highest signal frequency, 2*f c +B. In reality the sampling rate must be at least twice the highest information frequency, 2*B. Thus, the low-pass equivalent can be used in simulations. Low-pass equivalent signals are used to derive the complex envelope method, which is central to the simulation. The carrier modulated signal x(t) can be written as x(t) = r(t) cos( 2πf c t + φ(t) ) = Re( r(t) e j(2πfct+φ(t)) ) = Re( r(t) e jφ(t) e j2πfct ) r(t) is amplitude modulation, φ(t) is the phase modulation of the signal f c is the carrier frequency. The signal v(t) = r(t) e jφ(t) = p(t) + jq(t)
contains the information signal, and it is low-pass type of signal. It is often called the complex low-pass equivalent or the complex envelope of the signal. If the bandwidth of x(t) is B, and B << f c, then v(t) can be used to model the bandpass filtering of x(t). For simulation purposes the in-phase component of the bandpass signal is mapped into the real part of the complex envelope, and the quadrature component is mapped into the imaginary part. The real modulated signals have theoretically infinite bandwidth. In practice these signals can be thought to have an effective finite bandwidth, which makes the complex envelope method feasible. Sampling Sampling in simulation is a very important topic since it makes the interface between real and simulated systems. Of course it is a very important topic also in current real systems using AD-converters to convert analog signals into discrete world of DSP. The objective is to recover the response of a continuous system from a discrete simulation. By choosing the sampling rate properly this is possible for band-limited signals. Let x(t) be a band-limited function having a bandwidth f M. The representation of the signal x(t) in discrete form is x s (t) = x(t) s(t) where the impulse sequence s(t) = δ(t - nt s ) n = - is called the sampling function, T s is the sampling period, and f s = 1 / T s is the sampling frequency. If the sampling frequency f s > 2 f M then there is no overlap between the shifted replicas of the frequency spectrum, meaning that there is no aliasing. If f s < 2 f M then there is overlap causing aliasing. This is the sampling theorem stating as follows: Sampling theorem: a band-limited signal x(t) with X(f) = 0 for f f M is uniquely determined by its sample values x(nt s ) at a sequence of equidistant points t = nt s if f s > 2 f M, where f s = 1 / T s. The sampling frequency is known as the Nyquist rate. In reality some amount of aliasing is always present because practical signals are never strictly band-limited. When simulating non-linear systems the sampling rate comes even more important. The processing of a bandlimited baseband signal using a nonlinear system increases
the bandwidth, because the output a nonlinear system produces more frequency components than is put into it. Quite often the bandwidth of a nonlinear system is very large. There does exist some helpful things, though. First, the frequency components farther away from the center frequency have less energy than nearer components. This means that farther components give less contribution to the aliasing error. Also, generally the spectrum of x(t) is concentrated around the center frequency. The power series can be used to estimate the spectrum of a nonlinear system in order to select the correct sampling rate. Also a series of simulations using different sampling rates could be used. This might be a more realistic way to do it. The minimum sampling rate for simulation is, of cource, defined by the sampling theorem. In ideal system this is suitable. For simulations we have to use a much higher sampling rate. The higher sampling rate is needed to represent the time domain waveforms better and to model filters nonlinearities more accurately. For simulations the sampling rate can vary from 4 to 16 times the bandwidth. The needed sampling rate depends on the true S/N ratio needed. The aliased power must be well below the level of the true S/N. Interpolation and decimation These methods are for band-limited signals. The sampling rate and the signal bandwidth is changed in these operations. Interpolation is a method used to increase the sampling rate in a simulated system. This is done by adding zero-valued samples between previous samples, and filtering the result by using some low-pass filter. The signal bandwidth is increased in interpolation. Decimation is a method used to decrease the sampling rate. This is done by leaving some samples out between each of the taken samples. This is done together with a filtering operation, so some computational load can usually be saved. If the filter has the form of FIR-filter, there is no need to calculate the values of left-out samples, because their values are used for nothing. Only the shifting is needed. The signal bandwidth is decreased in decimation. A sampling rate alteration by a noninteger, but rational, factor can be done by cascading the operations of interpolation and decimation in that order to maintain the maximum possible bandwidth. In synchronization, for example, is usually needed a much larger sampling rate than in demodulation in order to get good bit error ratios.
Conclusions The sampling theorem states that the sampling frequency must be at least twice the highest information frequency. Also the bandlimited bandpass signals can be simulated using this theorem, because the complex envelope of the signal describes the contents of the bandpass signals quite well. The carrier frequency can be considered as a negligible term from the simulation point of view. When considering nonlinear systems the sampling rate must be chosen carefully. Nonlinearity may be simulated using a power series to estimate the spectrum. A more practical way is to simulate using different sampling rates. In real simulations the sampling rate should be much higher in order to avoid too much contribution to aliasing error. The suitable sampling rate is usually from 4 to 16 times the information bandwidth. Interpolation and decimation are used mainly to change the sampling rate for different purposes. Some functions need much higher sampling rates, but for those functions that do not need high sampling rate, using a high rate would be a waste of computer power and time. Interpolation increases the sampling rate and signal bandwidth, decimation decreases them. Also a cascade can be used so that decimation follows interpolation. Reference: Jeruchim, Balaban, Shanmugan: Simulation of Communication Systems, Plenum Press, 1992
Problem: Consider a communication system having channel spacing 25 khz (between adjacent channels) and effective bandwidth B = 19 khz. The samples are taken from second intermediate frequency f IM = 450 khz. The analog filter in second intermediate frequency is not sharp enough, so some amount of filtering of adjacent channel power is needed in digital domain. Question: Which one, 144 khz or 162 khz, is more suitable to be used as a sampling frequency? Explain why!