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. 2To 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. Cascad￾ing an anti-aliasing filter with a decimator gives a downsampler. OSB Figure 4.22 illustrates downsampling with and without aliasing. 2