Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341:DISCRETE-TIME SIGNAL PROCESSING OpenCourse Ware 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: xn] wn] xn xc(t) fi→f2 D/C C/D w回→ 入 万=1 f=2 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: xn]=xc(nT), Idln]=x[nM]=Ic(nMT). The discrete-time Fourier transform of x[n]and raln]are xe=-x((号-) xe=ax(”-)》
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] - f1 → f2 -w[n] ⇐⇒ x[n] - D/C - xc(t) C/D - w[n] 6 6 f1 = 1 T 1 f1 = 1 T 2 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(nMT). The discrete-time Fourier transform of x[n] and xd[n] are X(ejω) = 1 �∞ −∞ Xc � j �ω 2πk�� , T k= T − T 1 � ω 2πr �� Xd(ejω) = �∞ −∞ Xc � j . MT r= MT − MT 1
To relate X(ejw)and Xa(ej),rewrite with r=i+kM -o<k<o,0≤i≤M-1 一的-空三x(品警a) =X(ej(w-2mi)/M) 1 M-1 →Xa(e)= x(ei(w-2ri)/M). M =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 MT 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 xnj. x(n] lbsbua 1J111 ihlhiliel 0 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,and frequency scaling by.In general,the sampling rate of a signal can be reduced by a factor of M without aliasing if the signal is bandlimited to.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
To relate X(ejω) and Xd(ejω), rewrite with r = i + kM − ∞ < k < ∞, 0 ≤ i ≤ M − 1 1 M−1 � 1 ∞ � � ω 2πk 2πi ��� =⇒ Xd(ejω) = M � T � Xc j MT − T − MT i=0 k=−∞ j(ω−2πi)/M = X(e ) M−1 = Xd(ejω) = 1 � X(ej(ω−2πi)/M ⇒ ) . M 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 MT 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 2π domain, a decimator can be viewed as a sequence of two operations: replication at M , and 1 frequency scaling by M . In general, the sampling rate of a signal can be reduced by a factor of M without aliasing if the signal is bandlimited to π M . On the other hand, if the signal is not bandlimited, its bandwidth can be reduced first by discrete-time low pass filtering. Cascading 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(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. 3
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: ∞ 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
