Sampling Rate Conversion

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341: Discrete-Time Signal Processing OpenCourseWare 2006 Lecture 5 Sampling Rate Conversion Reading: Section 4.6 in Oppenheim, Schafer & Buck (OSB). It is often necessary to change the sampling rate of a discrete-time signal to obtain a new discrete-time representation of the underlying continuous-time signal. The desired system is shown below: x[n] - f →f 1 2 w[n] - ⇐⇒ x[n] - D/C xc (t) - C/D 6 f1 = 1 T1 w[n] - 6 f1 = 1 T2 Sample rate converter Sampling Rate Compression by an Integer Factor To reduce the sampling rate of a sequence by an integer factor, the sequence can be further compressed or decimated as depicted in OSB Figure 4.20. This discrete-time sampler can be interpreted as the cascade of a D/C converter and a C/D converter in which: x[n] = xc (nT ) , xd [n] = x[nM ] = xc (nM T ). The discrete-time Fourier transform of x[n] and xd [n] are � � �� 1 �∞ ω 2πk jω X(e ) = Xc j − , T k=−∞ T T �� � � 1 �∞ ω 2πr − X j . Xd (ejω ) = c M T r=−∞ MT MT 1 To relate X(ejω ) and Xd (ejω ), rewrite with r = i + kM − ∞ < k < ∞, 0 ≤ i ≤ M − 1 � � � ��� M −1 ∞ � � 1 1 ω 2πk 2πi Xc j =⇒ Xd (ejω ) = − − M T MT T MT i=0 k=−∞ = X(ej(ω−2πi)/M ) M −1 1 � X(ej(ω−2πi)/M ) . =⇒ Xd (e ) = M jω i=0 As an example, the following figure illustrates decimation by M = 2 in the time domain. We see that re-sampling the continuous signal at M T is equivalent to keeping only every M -th sample. In this cascaded system, the value of T is arbitrary and not affected by the original sampling frequency of x[n]. Time domain illustration of decimation at rate M = 2 OSB Figure 4.21 shows the corresponding frequency-domain representation. In the frequency domain, a decimator can be viewed as a sequence of two operations: replication at 2π M , and 1 . In general, the sampling rate of a signal can be reduced by a factor of frequency scaling by M π M without aliasing if the signal is bandlimited to M . On the other hand, if the signal is not bandlimited, its bandwidth can be reduced first by discrete-time low pass filtering. Cascad­ ing an anti-aliasing filter with a decimator gives a downsampler. OSB Figure 4.22 illustrates downsampling with and without aliasing. 2 Sampling Rate Expansion by an Integer Factor A typical system for increasing the sampling rate of a discrete sequence by an integer factor is illustrated in OSB Figure 4.24. Expressed in terms of Fourier transforms, the expander output is: Xe (ejω ) = = ∞ � n=−∞ ∞ � xe [n]e−jωn xe [k]e−jωkL = X(ejωL ) k=−∞ Expanding changes the time scale, and the LPF interpolates to fill in the missing values. As an example, the next figure shows upsampling at the rate of L = 2 in the time domain; for the corresponding spectra, see OSB Figure 4.25. Time domain illustration of upsampling at rate L = 2 Changing the Sampling Rate by a Non-Integer Factor By combining decimation and interpolation, the sampling rate of a sequence can be changed by a noninteger factor. For example, in OSB Figure 4.28 is a system for producing an output sequence with sampling period TLM . It is preferred that the interpolator precedes the decimator to avoid possible aliasing, ie. decimation first may create aliasing since the spectrum is replicated at less than 2π. By comparison, when a compressor and an expander are cascaded (without the LPF’s), it does not matter in what order they are placed, as long as the rates M and L are mutually prime. 3