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(eiu)= renle-jun n=-0 ∑ree-j =X(eiuL) k=-0 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. 00 x g(L=2) 年 ● wIn] 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 TM.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 2T.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. 3Sampling 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: ∞ e X −jωn e(ejω) = � xe[n] n=−∞ ∞ e−jωkL jωL = ) � xe[k] = X(e 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 TM . It is preferred that the interpolator precedes the decimator L 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