SHANNONS INFORMATION THEOREM AND NYQUISTS CRITERIA Shannon: An Analog Signal with a Bandwidth of fa Must be Sampled at a Rate of f 2fa in Order to Avoid the Loss of Information The signal bandwidth may extend from DC to fa(Baseband Sampling)or from f, to f2, where fa=f2-f,(Undersampling, Bandpass Sampling, Harmonic Sampling, Super-Nyquist) Nyquist: If f <2fa, then a Phenomena Called Aliasing Will Occur. Aliasing is used to advantage in undersampling applications Figure 5.2 In order to understand the implications of aliasing in both the time and frequency domain, first consider the four cases of a time domain representation of a sampled sinewave signal shown in Figure 5.3. In Case 1, it is clear that an adequate number of samples have been taken to preserve the information about the sinewave In Case 2 of the figure, only four samples per cycle are taken; still an adequate number to preserve the information Case 3 represents the ambiguous limiting condition where fs=2fa. If the relationship between the sampling points and the sinewave were such that the sinewave was being sampled at precisely the zero crossings (rather than at the peaks, as shown in the illustration), then all information regarding the sinewave would be lost. Case 4 of Figure 5.3 represents the situation where fs<2fa, and th information obtained from the samples indicates a sinewave having a frequency which is lower than fs/2, i.e. the out-of-band signal is aliased into the Nyquist bandwidth between dc and f/2. As the sampling rate is further decreased, and the analog input frequency fa approaches the sampling frequency fs, the aliased signal approaches dc in the frequency spectrum3 SHANNON’S INFORMATION THEOREM AND NYQUIST’S CRITERIA Shannon: An Analog Signal with a Bandwidth of fa Must be Sampled at a Rate of fs>2fa in Order to Avoid the Loss of Information. The signal bandwidth may extend from DC to fa (Baseband Sampling) or from f1 to f2 , where fa = f2 - f1 (Undersampling, Bandpass Sampling, Harmonic Sampling, Super-Nyquist) Nyquist: If fs<2fa , then a Phenomena Called Aliasing Will Occur. Aliasing is used to advantage in undersampling applications. Figure 5.2 In order to understand the implications of aliasing in both the time and frequency domain, first consider the four cases of a time domain representation of a sampled sinewave signal shown in Figure 5.3. In Case 1, it is clear that an adequate number of samples have been taken to preserve the information about the sinewave. In Case 2 of the figure, only four samples per cycle are taken; still an adequate number to preserve the information. Case 3 represents the ambiguous limiting condition where fs=2fa. If the relationship between the sampling points and the sinewave were such that the sinewave was being sampled at precisely the zero crossings (rather than at the peaks, as shown in the illustration), then all information regarding the sinewave would be lost. Case 4 of Figure 5.3 represents the situation where fs<2fa, and the information obtained from the samples indicates a sinewave having a frequency which is lower than fs /2, i.e. the out-of -band signal is aliased into the Nyquist bandwidth between dc and fs /2. As the sampling rate is further decreased, and the analog input frequency fa approaches the sampling frequency fs , the aliased signal approaches dc in the frequency spectrum