Lecture 13: Pulse Code Modulation, Study notes of Law

Download Lecture 13: Pulse Code Modulation and more Study notes Law in PDF only on Docsity! Lecture 13: Pulse Code Modulation John M Pauly November 2, 2021 Pulse Code Modulation (PCM) I PCM in the Bell System I Multiplexing PCM I Asynchronous PCM I Extensions to PCM I Differential PCM (DPCM) I Adaptive DPCM (ADPCM) I Delta-Sigma Modulation (DM) I Vocoders Based on lecture notes from John Gill Electromechanical Crossbar Switch T-1 Carrier System The input to the (fast) 13-bit ADC comes from an analog multiplexer. The digital processor compresses the digital value according to µ-law. T-1 Carrier System (cont.) The 8-bit compressed voice values are sent consecutively, msb first. The samples of all 24 inputs comprise a frame. Most serial communications transmits data lsb first (“little endian”). Multiplexing PCM A major motivation for PCM is the ability to multiplex many low bit rate channels on a single hit bit rate channel. There are many ways to do this: I Bit interleaving I Word interleaving These each have advantages and disadvantages. A major issue is synchronization. There are several different approaches I Synchronous : hard to do in practice I Asynchronous : potentially wasteful of capacity I Plesiochronous : a practical balance Multiplexing PCM Amy Bit Level Multiplexing A — AgByCyD4ApByCoDo. 5 s Gn —_ FUL pany oa > Te aL —_1 aoa Word Level Multiplexing Asynchronous PCM Problem: It is difficult to ensure that the bits arrive and leave at the synchronous rates Example : I 100 km cable, carrying 200 Mbits/s. I If the temperature increases by 1 ◦F, it increases the propagation velocity by 0.01%. I This results in a temporary increase of 20 kbits/s in the bit arrival rate. What do we do with all the extra bits? Answers : I Run the link at a slightly slower bit rate, and bit stuff the extra bits empty bits if you don’t need them I Run the link at a slightly lower bit rate, and drop occasional LSB’s I Run at the ideal rate, and bit stuff/bit delete as needed Use the control channel to indicate which bits are stuffed/deleted Differential PCM We’ll start with the receiver + Predictor + + Decoder <latexit sha1_base64="xsjev+pcrDIaa8pjijqUoS6f8a4=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKewGUY8BLx4jmAdsljA7mU3GzGOdmRVCyD948aCIV//Hm3/jJNmDJhY0FFXddHfFKWfG+v63V1hb39jcKm6Xdnb39g/Kh0ctozJNaJMornQnxoZyJmnTMstpJ9UUi5jTdjy6mfntJ6oNU/LejlMaCTyQLGEEWye1RO8xHEW9csWv+nOgVRLkpAI5Gr3yV7evSCaotIRjY8LAT200wdoywum01M0MTTEZ4QENHZVYUBNN5tdO0ZlT+ihR2pW0aK7+nphgYcxYxK5TYDs0y95M/M8LM5tcRxMm08xSSRaLkowjq9DsddRnmhLLx45gopm7FZEh1phYF1DJhRAsv7xKWrVqcFkN7i4q9VoeRxFO4BTOIYArqMMtNKAJBB7gGV7hzVPei/fufSxaC14+cwx/4H3+AIXWjww=</latexit> mq[k] <latexit sha1_base64="S/MVswN104LVFyQ3bbijvXL0y5Q=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkR9Vjw4rGC/YAklM122y7dTeLuRCihf8OLB0W8+me8+W/ctjlo64OBx3szzMyLUikMuu63s7a+sbm1Xdop7+7tHxxWjo7bJsk04y2WyER3I2q4FDFvoUDJu6nmVEWSd6Lx7czvPHFtRBI/4CTloaLDWAwEo2ilIBhRzNW09+iPw16l6tbcOcgq8QpShQLNXuUr6CcsUzxGJqkxvuemGOZUo2CST8tBZnhK2ZgOuW9pTBU3YT6/eUrOrdIng0TbipHM1d8TOVXGTFRkOxXFkVn2ZuJ/np/h4CbMRZxmyGO2WDTIJMGEzAIgfaE5QzmxhDIt7K2EjaimDG1MZRuCt/zyKmnXa95Vzbu/rDbqRRwlOIUzuAAPrqEBd9CEFjBI4Rle4c3JnBfn3flYtK45xcwJ/IHz+QNc35HZ</latexit> m̂q[k] <latexit sha1_base64="U6P7OMGWLS7HVxin2mSblHvYpEw=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWw2m3btZjfuboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MOVMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WmCG0TyaXqhVhTzgRtG2Y47aWK4iTktBuOb2Z+94kqzaS4N5OUBgkeChYzgo2VOtHg0R8Hg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY619z01NkGNlGOF0WulnmqaYjPGQ+pYKnFAd5PNrp+jMKhGKpbIlDJqrvydynGg9SULbmWAz0sveTPzP8zMTXwc5E2lmqCCLRXHGkZFo9jqKmKLE8IklmChmb0VkhBUmxgZUsSF4yy+vkk6j7l3WvbuLWrNRxFGGEziFc/DgCppwCy1oA4EHeIZXeHOk8+K8Ox+L1pJTzBzDHzifP3gFjwM=</latexit> dq[k] I We’ll assume that the system is working, and has produced previous values of mq[k] I The predictor compute m̂q[k], the next predicted sample I The input is the quantized difference signal dq[k] I This is added to the predicted signal m̂q[k] to recover mq[k]. Differential PCM A simple encoder would quantize first, compute the prediction, and then compute the difference signal to transmit + Quantizer Predictor + - <latexit sha1_base64="aO789MA/gNUEOlj3CoM5Fvh4NIc=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWy2m3bp7ibsboQS+he8eFDEq3/Im//GTZqDtj4YeLw3w8y8MOFMG9f9diobm1vbO9Xd2t7+weFR/fikp+NUEdolMY/VIMSaciZp1zDD6SBRFIuQ0344u8v9/hNVmsXy0cwTGgg8kSxiBJtcEv4sGNUbbtMtgNaJV5IGlOiM6l/DcUxSQaUhHGvte25iggwrwwini9ow1TTBZIYn1LdUYkF1kBW3LtCFVcYoipUtaVCh/p7IsNB6LkLbKbCZ6lUvF//z/NREt0HGZJIaKslyUZRyZGKUP47GTFFi+NwSTBSztyIyxQoTY+Op2RC81ZfXSa/V9K6b3sNVo90q46jCGZzDJXhwA224hw50gcAUnuEV3hzhvDjvzseyteKUM6fwB87nD/z5jig=</latexit> m[k] <latexit sha1_base64="U6P7OMGWLS7HVxin2mSblHvYpEw=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWw2m3btZjfuboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MOVMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WmCG0TyaXqhVhTzgRtG2Y47aWK4iTktBuOb2Z+94kqzaS4N5OUBgkeChYzgo2VOtHg0R8Hg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY619z01NkGNlGOF0WulnmqaYjPGQ+pYKnFAd5PNrp+jMKhGKpbIlDJqrvydynGg9SULbmWAz0sveTPzP8zMTXwc5E2lmqCCLRXHGkZFo9jqKmKLE8IklmChmb0VkhBUmxgZUsSF4yy+vkk6j7l3WvbuLWrNRxFGGEziFc/DgCppwCy1oA4EHeIZXeHOk8+K8Ox+L1pJTzBzDHzifP3gFjwM=</latexit> dq[k] <latexit sha1_base64="S/MVswN104LVFyQ3bbijvXL0y5Q=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkR9Vjw4rGC/YAklM122y7dTeLuRCihf8OLB0W8+me8+W/ctjlo64OBx3szzMyLUikMuu63s7a+sbm1Xdop7+7tHxxWjo7bJsk04y2WyER3I2q4FDFvoUDJu6nmVEWSd6Lx7czvPHFtRBI/4CTloaLDWAwEo2ilIBhRzNW09+iPw16l6tbcOcgq8QpShQLNXuUr6CcsUzxGJqkxvuemGOZUo2CST8tBZnhK2ZgOuW9pTBU3YT6/eUrOrdIng0TbipHM1d8TOVXGTFRkOxXFkVn2ZuJ/np/h4CbMRZxmyGO2WDTIJMGEzAIgfaE5QzmxhDIt7K2EjaimDG1MZRuCt/zyKmnXa95Vzbu/rDbqRRwlOIUzuAAPrqEBd9CEFjBI4Rle4c3JnBfn3flYtK45xcwJ/IHz+QNc35HZ</latexit> m̂q[k] <latexit sha1_base64="xsjev+pcrDIaa8pjijqUoS6f8a4=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKewGUY8BLx4jmAdsljA7mU3GzGOdmRVCyD948aCIV//Hm3/jJNmDJhY0FFXddHfFKWfG+v63V1hb39jcKm6Xdnb39g/Kh0ctozJNaJMornQnxoZyJmnTMstpJ9UUi5jTdjy6mfntJ6oNU/LejlMaCTyQLGEEWye1RO8xHEW9csWv+nOgVRLkpAI5Gr3yV7evSCaotIRjY8LAT200wdoywum01M0MTTEZ4QENHZVYUBNN5tdO0ZlT+ihR2pW0aK7+nphgYcxYxK5TYDs0y95M/M8LM5tcRxMm08xSSRaLkowjq9DsddRnmhLLx45gopm7FZEh1phYF1DJhRAsv7xKWrVqcFkN7i4q9VoeRxFO4BTOIYArqMMtNKAJBB7gGV7hzVPei/fufSxaC14+cwx/4H3+AIXWjww=</latexit> mq[k] I The problem with this is that quantization errors can build up I This can lead to larger errors. I We can do much better! Differential PCM However, we can do better with this encoder + Quantizer + Predictor + - ++ Decoder Encoder <latexit sha1_base64="xsjev+pcrDIaa8pjijqUoS6f8a4=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKewGUY8BLx4jmAdsljA7mU3GzGOdmRVCyD948aCIV//Hm3/jJNmDJhY0FFXddHfFKWfG+v63V1hb39jcKm6Xdnb39g/Kh0ctozJNaJMornQnxoZyJmnTMstpJ9UUi5jTdjy6mfntJ6oNU/LejlMaCTyQLGEEWye1RO8xHEW9csWv+nOgVRLkpAI5Gr3yV7evSCaotIRjY8LAT200wdoywum01M0MTTEZ4QENHZVYUBNN5tdO0ZlT+ihR2pW0aK7+nphgYcxYxK5TYDs0y95M/M8LM5tcRxMm08xSSRaLkowjq9DsddRnmhLLx45gopm7FZEh1phYF1DJhRAsv7xKWrVqcFkN7i4q9VoeRxFO4BTOIYArqMMtNKAJBB7gGV7hzVPei/fufSxaC14+cwx/4H3+AIXWjww=</latexit> mq[k] <latexit sha1_base64="aO789MA/gNUEOlj3CoM5Fvh4NIc=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWy2m3bp7ibsboQS+he8eFDEq3/Im//GTZqDtj4YeLw3w8y8MOFMG9f9diobm1vbO9Xd2t7+weFR/fikp+NUEdolMY/VIMSaciZp1zDD6SBRFIuQ0344u8v9/hNVmsXy0cwTGgg8kSxiBJtcEv4sGNUbbtMtgNaJV5IGlOiM6l/DcUxSQaUhHGvte25iggwrwwini9ow1TTBZIYn1LdUYkF1kBW3LtCFVcYoipUtaVCh/p7IsNB6LkLbKbCZ6lUvF//z/NREt0HGZJIaKslyUZRyZGKUP47GTFFi+NwSTBSztyIyxQoTY+Op2RC81ZfXSa/V9K6b3sNVo90q46jCGZzDJXhwA224hw50gcAUnuEV3hzhvDjvzseyteKUM6fwB87nD/z5jig=</latexit> m[k] <latexit sha1_base64="42gXfy6wP0BckHyvAUkhX/w8mgw=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWw2m3bp7ibsboQS+he8eFDEq3/Im//GTZuDtj4YeLw3w8y8MOVMG9f9diobm1vbO9Xd2t7+weFR/fikp5NMEdolCU/UIMSaciZp1zDD6SBVFIuQ0344vSv8/hNVmiXy0cxSGgg8lixmBJtCivxpMKo33Ka7AFonXkkaUKIzqn8No4RkgkpDONba99zUBDlWhhFO57VhpmmKyRSPqW+pxILqIF/cOkcXVolQnChb0qCF+nsix0LrmQhtp8Bmole9QvzP8zMT3wY5k2lmqCTLRXHGkUlQ8TiKmKLE8JklmChmb0VkghUmxsZTsyF4qy+vk16r6V03vYerRrtVxlGFMziHS/DgBtpwDx3oAoEJPMMrvDnCeXHenY9la8UpZ07hD5zPH+86jh8=</latexit> d[k] <latexit sha1_base64="U6P7OMGWLS7HVxin2mSblHvYpEw=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWw2m3btZjfuboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MOVMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WmCG0TyaXqhVhTzgRtG2Y47aWK4iTktBuOb2Z+94kqzaS4N5OUBgkeChYzgo2VOtHg0R8Hg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY619z01NkGNlGOF0WulnmqaYjPGQ+pYKnFAd5PNrp+jMKhGKpbIlDJqrvydynGg9SULbmWAz0sveTPzP8zMTXwc5E2lmqCCLRXHGkZFo9jqKmKLE8IklmChmb0VkhBUmxgZUsSF4yy+vkk6j7l3WvbuLWrNRxFGGEziFc/DgCppwCy1oA4EHeIZXeHOk8+K8Ox+L1pJTzBzDHzifP3gFjwM=</latexit> dq[k] <latexit sha1_base64="S/MVswN104LVFyQ3bbijvXL0y5Q=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkR9Vjw4rGC/YAklM122y7dTeLuRCihf8OLB0W8+me8+W/ctjlo64OBx3szzMyLUikMuu63s7a+sbm1Xdop7+7tHxxWjo7bJsk04y2WyER3I2q4FDFvoUDJu6nmVEWSd6Lx7czvPHFtRBI/4CTloaLDWAwEo2ilIBhRzNW09+iPw16l6tbcOcgq8QpShQLNXuUr6CcsUzxGJqkxvuemGOZUo2CST8tBZnhK2ZgOuW9pTBU3YT6/eUrOrdIng0TbipHM1d8TOVXGTFRkOxXFkVn2ZuJ/np/h4CbMRZxmyGO2WDTIJMGEzAIgfaE5QzmxhDIt7K2EjaimDG1MZRuCt/zyKmnXa95Vzbu/rDbqRRwlOIUzuAAPrqEBd9CEFjBI4Rle4c3JnBfn3flYtK45xcwJ/IHz+QNc35HZ</latexit> m̂q[k] I This uses the actual decoder to compute the prediction I It computes the difference between the unquantized signal and the quantized prediction. Delta Modulation DPCM is much simpler if we just use the previous sample for the predictor. The predictor is just a delay. + Delay + + Decoder <latexit sha1_base64="xsjev+pcrDIaa8pjijqUoS6f8a4=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKewGUY8BLx4jmAdsljA7mU3GzGOdmRVCyD948aCIV//Hm3/jJNmDJhY0FFXddHfFKWfG+v63V1hb39jcKm6Xdnb39g/Kh0ctozJNaJMornQnxoZyJmnTMstpJ9UUi5jTdjy6mfntJ6oNU/LejlMaCTyQLGEEWye1RO8xHEW9csWv+nOgVRLkpAI5Gr3yV7evSCaotIRjY8LAT200wdoywum01M0MTTEZ4QENHZVYUBNN5tdO0ZlT+ihR2pW0aK7+nphgYcxYxK5TYDs0y95M/M8LM5tcRxMm08xSSRaLkowjq9DsddRnmhLLx45gopm7FZEh1phYF1DJhRAsv7xKWrVqcFkN7i4q9VoeRxFO4BTOIYArqMMtNKAJBB7gGV7hzVPei/fufSxaC14+cwx/4H3+AIXWjww=</latexit> mq[k] <latexit sha1_base64="U6P7OMGWLS7HVxin2mSblHvYpEw=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWw2m3btZjfuboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MOVMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WmCG0TyaXqhVhTzgRtG2Y47aWK4iTktBuOb2Z+94kqzaS4N5OUBgkeChYzgo2VOtHg0R8Hg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY619z01NkGNlGOF0WulnmqaYjPGQ+pYKnFAd5PNrp+jMKhGKpbIlDJqrvydynGg9SULbmWAz0sveTPzP8zMTXwc5E2lmqCCLRXHGkZFo9jqKmKLE8IklmChmb0VkhBUmxgZUsSF4yy+vkk6j7l3WvbuLWrNRxFGGEziFc/DgCppwCy1oA4EHeIZXeHOk8+K8Ox+L1pJTzBzDHzifP3gFjwM=</latexit> dq[k] <latexit sha1_base64="/WILumvugzsGT3HCiU0k4YpleJM=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRbBi2W3FPVY8OKxgv2A7VKyabYNTbLbJCuUpX/CiwdFvPp3vPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFaEtEvNYdUOsKWeStgwznHYTRbEIOe2E47u533miSrNYPpppQgOBh5JFjGBjpa7oT/zxlRf0yxW36i6A1omXkwrkaPbLX71BTFJBpSEca+17bmKCDCvDCKezUi/VNMFkjIfUt1RiQXWQLe6doQurDFAUK1vSoIX6eyLDQuupCG2nwGakV725+J/npya6DTImk9RQSZaLopQjE6P582jAFCWGTy3BRDF7KyIjrDAxNqKSDcFbfXmdtGtV77rqPdQrjVoeRxHO4BwuwYMbaMA9NKEFBDg8wyu8ORPnxXl3PpatBSefOYU/cD5/AGLMj34=</latexit> mq[k 1] + Quantizer + Delay + - ++ Decoder Encoder <latexit sha1_base64="xsjev+pcrDIaa8pjijqUoS6f8a4=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKewGUY8BLx4jmAdsljA7mU3GzGOdmRVCyD948aCIV//Hm3/jJNmDJhY0FFXddHfFKWfG+v63V1hb39jcKm6Xdnb39g/Kh0ctozJNaJMornQnxoZyJmnTMstpJ9UUi5jTdjy6mfntJ6oNU/LejlMaCTyQLGEEWye1RO8xHEW9csWv+nOgVRLkpAI5Gr3yV7evSCaotIRjY8LAT200wdoywum01M0MTTEZ4QENHZVYUBNN5tdO0ZlT+ihR2pW0aK7+nphgYcxYxK5TYDs0y95M/M8LM5tcRxMm08xSSRaLkowjq9DsddRnmhLLx45gopm7FZEh1phYF1DJhRAsv7xKWrVqcFkN7i4q9VoeRxFO4BTOIYArqMMtNKAJBB7gGV7hzVPei/fufSxaC14+cwx/4H3+AIXWjww=</latexit> mq[k] <latexit sha1_base64="aO789MA/gNUEOlj3CoM5Fvh4NIc=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWy2m3bp7ibsboQS+he8eFDEq3/Im//GTZqDtj4YeLw3w8y8MOFMG9f9diobm1vbO9Xd2t7+weFR/fikp+NUEdolMY/VIMSaciZp1zDD6SBRFIuQ0344u8v9/hNVmsXy0cwTGgg8kSxiBJtcEv4sGNUbbtMtgNaJV5IGlOiM6l/DcUxSQaUhHGvte25iggwrwwini9ow1TTBZIYn1LdUYkF1kBW3LtCFVcYoipUtaVCh/p7IsNB6LkLbKbCZ6lUvF//z/NREt0HGZJIaKslyUZRyZGKUP47GTFFi+NwSTBSztyIyxQoTY+Op2RC81ZfXSa/V9K6b3sNVo90q46jCGZzDJXhwA224hw50gcAUnuEV3hzhvDjvzseyteKUM6fwB87nD/z5jig=</latexit> m[k] <latexit sha1_base64="42gXfy6wP0BckHyvAUkhX/w8mgw=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWw2m3bp7ibsboQS+he8eFDEq3/Im//GTZuDtj4YeLw3w8y8MOVMG9f9diobm1vbO9Xd2t7+weFR/fikp5NMEdolCU/UIMSaciZp1zDD6SBVFIuQ0344vSv8/hNVmiXy0cxSGgg8lixmBJtCivxpMKo33Ka7AFonXkkaUKIzqn8No4RkgkpDONba99zUBDlWhhFO57VhpmmKyRSPqW+pxILqIF/cOkcXVolQnChb0qCF+nsix0LrmQhtp8Bmole9QvzP8zMT3wY5k2lmqCTLRXHGkUlQ8TiKmKLE8JklmChmb0VkghUmxsZTsyF4qy+vk16r6V03vYerRrtVxlGFMziHS/DgBtpwDx3oAoEJPMMrvDnCeXHenY9la8UpZ07hD5zPH+86jh8=</latexit> d[k] <latexit sha1_base64="U6P7OMGWLS7HVxin2mSblHvYpEw=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWw2m3btZjfuboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MOVMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WmCG0TyaXqhVhTzgRtG2Y47aWK4iTktBuOb2Z+94kqzaS4N5OUBgkeChYzgo2VOtHg0R8Hg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY619z01NkGNlGOF0WulnmqaYjPGQ+pYKnFAd5PNrp+jMKhGKpbIlDJqrvydynGg9SULbmWAz0sveTPzP8zMTXwc5E2lmqCCLRXHGkZFo9jqKmKLE8IklmChmb0VkhBUmxgZUsSF4yy+vkk6j7l3WvbuLWrNRxFGGEziFc/DgCppwCy1oA4EHeIZXeHOk8+K8Ox+L1pJTzBzDHzifP3gFjwM=</latexit> dq[k] <latexit sha1_base64="/WILumvugzsGT3HCiU0k4YpleJM=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRbBi2W3FPVY8OKxgv2A7VKyabYNTbLbJCuUpX/CiwdFvPp3vPlvTNs9aOuDgcd7M8zMCxPOtHHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFaEtEvNYdUOsKWeStgwznHYTRbEIOe2E47u533miSrNYPpppQgOBh5JFjGBjpa7oT/zxlRf0yxW36i6A1omXkwrkaPbLX71BTFJBpSEca+17bmKCDCvDCKezUi/VNMFkjIfUt1RiQXWQLe6doQurDFAUK1vSoIX6eyLDQuupCG2nwGakV725+J/npya6DTImk9RQSZaLopQjE6P582jAFCWGTy3BRDF7KyIjrDAxNqKSDcFbfXmdtGtV77rqPdQrjVoeRxHO4BwuwYMbaMA9NKEFBDg8wyu8ORPnxXl3PpatBSefOYU/cD5/AGLMj34=</latexit> mq[k 1] We can make up for the simple predictor by increasing the sampling rate. This increases the similarity between adjacent samples, and reduces the number of bits we need for dq[k]. Sigma-Delta Modulation If the predictor for DPCM is sufficiently accurate, we only need one bit for the error. The waveform we need to transmit 0, 1, and -1, I The output waveform is simple the sum of the transmitted errors. Fidelity is limited by the derivative of the signal. I We can improve fidelity with faster sampling rate, so that adjacent samples are highly correlated. I Typical numbers might be 8 times the Nyquist rate, or more. Sigma-Delta Modulation The encoder and decoder are particularly simple + + + Encoder <latexit sha1_base64="6m8YMn19bqsnU+S1OVeXxpJLPws=">AAAB63icbVBNS8NAEJ3Ur1q/qh69BItQLyUpoh4LXjxWsB/QhrLZbtqlu5uwOxFK6V/w4kERr/4hb/4bN20O2vpg4PHeDDPzwkRwg5737RQ2Nre2d4q7pb39g8Oj8vFJ28SppqxFYxHrbkgME1yxFnIUrJtoRmQoWCec3GV+54lpw2P1iNOEBZKMFI84JZhJsoqXg3LFq3kLuOvEz0kFcjQH5a/+MKapZAqpIMb0fC/BYEY0cirYvNRPDUsInZAR61mqiGQmmC1unbsXVhm6UaxtKXQX6u+JGZHGTGVoOyXBsVn1MvE/r5didBvMuEpSZIouF0WpcDF2s8fdIdeMophaQqjm9laXjokmFG08JRuCv/ryOmnXa/51zX+4qjTqeRxFOINzqIIPN9CAe2hCCyiM4Rle4c2Rzovz7nwsWwtOPnMKf+B8/gBuJI3K</latexit> m(t) <latexit sha1_base64="7NdMPQf+GAXXprThKTCO0/1zsD4=">AAAB63icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E1DJgYxnBxEByhL3NJlmyu3fszgnhyF+wsVDE1j9k579xL7lCEx8MPN6bYWZelEhh0fe/vdLa+sbmVnm7srO7t39QPTxq2zg1jLdYLGPTiajlUmjeQoGSdxLDqYokf4wmt7n/+MSNFbF+wGnCQ0VHWgwFo5hLPaGxX635dX8OskqCgtSgQLNf/eoNYpYqrpFJam038BMMM2pQMMlnlV5qeULZhI5411FNFbdhNr91Rs6cMiDD2LjSSObq74mMKmunKnKdiuLYLnu5+J/XTXF4E2ZCJylyzRaLhqkkGJP8cTIQhjOUU0coM8LdStiYGsrQxVNxIQTLL6+S9kU9uKoH95e1hl/EUYYTOIVzCOAaGnAHTWgBgzE8wyu8ecp78d69j0VryStmjuEPvM8fHzaOPQ==</latexit> <latexit sha1_base64="x4X3d7kP2qj/aVawHPK9MnAyHpc=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBahXkpSRD0WvHisYFuxDWWz3bRLN5uwOxFK6L/w4kERr/4bb/4bt20O2vpg4PHeDDPzgkQKg6777RTW1jc2t4rbpZ3dvf2D8uFR28SpZrzFYhnrh4AaLoXiLRQo+UOiOY0CyTvB+Gbmd564NiJW9zhJuB/RoRKhYBSt9NgbUcyiaRXP++WKW3PnIKvEy0kFcjT75a/eIGZpxBUySY3pem6CfkY1Cib5tNRLDU8oG9Mh71qqaMSNn80vnpIzqwxIGGtbCslc/T2R0ciYSRTYzojiyCx7M/E/r5tieO1nQiUpcsUWi8JUEozJ7H0yEJozlBNLKNPC3krYiGrK0IZUsiF4yy+vkna95l3WvLuLSqOex1GEEziFKnhwBQ24hSa0gIGCZ3iFN8c4L86787FoLTj5zDH8gfP5Az/zkJc=</latexit> m̂(t) <latexit sha1_base64="7NdMPQf+GAXXprThKTCO0/1zsD4=">AAAB63icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E1DJgYxnBxEByhL3NJlmyu3fszgnhyF+wsVDE1j9k579xL7lCEx8MPN6bYWZelEhh0fe/vdLa+sbmVnm7srO7t39QPTxq2zg1jLdYLGPTiajlUmjeQoGSdxLDqYokf4wmt7n/+MSNFbF+wGnCQ0VHWgwFo5hLPaGxX635dX8OskqCgtSgQLNf/eoNYpYqrpFJam038BMMM2pQMMlnlV5qeULZhI5411FNFbdhNr91Rs6cMiDD2LjSSObq74mMKmunKnKdiuLYLnu5+J/XTXF4E2ZCJylyzRaLhqkkGJP8cTIQhjOUU0coM8LdStiYGsrQxVNxIQTLL6+S9kU9uKoH95e1hl/EUYYTOIVzCOAaGnAHTWgBgzE8wyu8ecp78d69j0VryStmjuEPvM8fHzaOPQ==</latexit> <latexit sha1_base64="U6P7OMGWLS7HVxin2mSblHvYpEw=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWw2m3btZjfuboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MOVMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WmCG0TyaXqhVhTzgRtG2Y47aWK4iTktBuOb2Z+94kqzaS4N5OUBgkeChYzgo2VOtHg0R8Hg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY619z01NkGNlGOF0WulnmqaYjPGQ+pYKnFAd5PNrp+jMKhGKpbIlDJqrvydynGg9SULbmWAz0sveTPzP8zMTXwc5E2lmqCCLRXHGkZFo9jqKmKLE8IklmChmb0VkhBUmxgZUsSF4yy+vkk6j7l3WvbuLWrNRxFGGEziFc/DgCppwCy1oA4EHeIZXeHOk8+K8Ox+L1pJTzBzDHzifP3gFjwM=</latexit> dq[k] Integrator Lowpass Integrator Comparitor <latexit sha1_base64="6m8YMn19bqsnU+S1OVeXxpJLPws=">AAAB63icbVBNS8NAEJ3Ur1q/qh69BItQLyUpoh4LXjxWsB/QhrLZbtqlu5uwOxFK6V/w4kERr/4hb/4bN20O2vpg4PHeDDPzwkRwg5737RQ2Nre2d4q7pb39g8Oj8vFJ28SppqxFYxHrbkgME1yxFnIUrJtoRmQoWCec3GV+54lpw2P1iNOEBZKMFI84JZhJsoqXg3LFq3kLuOvEz0kFcjQH5a/+MKapZAqpIMb0fC/BYEY0cirYvNRPDUsInZAR61mqiGQmmC1unbsXVhm6UaxtKXQX6u+JGZHGTGVoOyXBsVn1MvE/r5didBvMuEpSZIouF0WpcDF2s8fdIdeMophaQqjm9laXjokmFG08JRuCv/ryOmnXa/51zX+4qjTqeRxFOINzqIIPN9CAe2hCCyiM4Rle4c2Rzovz7nwsWwtOPnMKf+B8/gBuJI3K</latexit> m(t) <latexit sha1_base64="U6P7OMGWLS7HVxin2mSblHvYpEw=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWw2m3btZjfuboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MOVMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WmCG0TyaXqhVhTzgRtG2Y47aWK4iTktBuOb2Z+94kqzaS4N5OUBgkeChYzgo2VOtHg0R8Hg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY619z01NkGNlGOF0WulnmqaYjPGQ+pYKnFAd5PNrp+jMKhGKpbIlDJqrvydynGg9SULbmWAz0sveTPzP8zMTXwc5E2lmqCCLRXHGkZFo9jqKmKLE8IklmChmb0VkhBUmxgZUsSF4yy+vkk6j7l3WvbuLWrNRxFGGEziFc/DgCppwCy1oA4EHeIZXeHOk8+K8Ox+L1pJTzBzDHzifP3gFjwM=</latexit> dq[k] Decoder <latexit sha1_base64="hmjPiQR80jOscEOHjtiUlQIphuw=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkpSRD0WvHisYFuhCWWz3bZLd5O4OxFK6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzwkQKg6777RTW1jc2t4rbpZ3dvf2D8uFR28SpZrzFYhnrh5AaLkXEWyhQ8odEc6pCyTvh+Gbmd564NiKO7nGS8EDRYSQGglG0ku+PKGZq2nus4nmvXHFr7hxklXg5qUCOZq/85fdjlioeIZPUmK7nJhhkVKNgkk9Lfmp4QtmYDnnX0ogqboJsfvOUnFmlTwaxthUhmau/JzKqjJmo0HYqiiOz7M3E/7xuioPrIBNRkiKP2GLRIJUEYzILgPSF5gzlxBLKtLC3EjaimjK0MZVsCN7yy6ukXa95lzXv7qLSqOdxFOEETqEKHlxBA26hCS1gkMAzvMKbkzovzrvzsWgtOPnMMfyB8/kDzfuRew==</latexit> m̂q(t) <latexit sha1_base64="U6P7OMGWLS7HVxin2mSblHvYpEw=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeCF48V7AekoWw2m3btZjfuboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MOVMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WmCG0TyaXqhVhTzgRtG2Y47aWK4iTktBuOb2Z+94kqzaS4N5OUBgkeChYzgo2VOtHg0R8Hg2rNrbtzoFXiFaQGBVqD6lc/kiRLqDCEY619z01NkGNlGOF0WulnmqaYjPGQ+pYKnFAd5PNrp+jMKhGKpbIlDJqrvydynGg9SULbmWAz0sveTPzP8zMTXwc5E2lmqCCLRXHGkZFo9jqKmKLE8IklmChmb0VkhBUmxgZUsSF4yy+vkk6j7l3WvbuLWrNRxFGGEziFc/DgCppwCy1oA4EHeIZXeHOk8+K8Ox+L1pJTzBzDHzifP3gFjwM=</latexit> dq[k] I Result can be very high fidelity if the sampling rate is high enough I Widely used in consumer electronics, and elsewhere I This is why you see ”1 bit A/D” stickers on products, and it is actually a good thing. I Similar number of bits per second, but much simpler hardware