COMPANDORS ON NARROW BAND SPEECH CHANNELS 1217 



although any broadband signal can be similarly folded into a 4 kc band by 

 an instantaneous sampler with an 8 kc repetition rate, the process is not 

 fully reversible. For example, there is no means of telling whether a 3 kc 

 component in the folded signal comes from a 3 kc, a 5 kc, an 11 kc, or a 13 

 kc, etc. component in the original signal. Hence it is only a very special 

 class of signals that can be recovered after their frequency spectra have been 

 condensed in this fashion. 



To recover the original speech in this case, the spectrum shown in Fig. 

 2(a) can be sampled at an 8 kc rate to produce the spectrum shown in Fig. 

 2(b). Now an examination of the spectra involved will show that when the 

 second sampling is properly synchronized with the transmitting sampler, the 

 two spectra shown in 1(d) and 2(b) will be identical. The spectrum in Fig. 

 1(d) represents the 8000 samples per second of the compressed speech gen- 

 erated at the transmitter. Thus, when the spectra of Figs. 1(d) and 2(b) are 

 identical, samples will be recovered at the receiver which are identical to 

 those that were generated at the transmitter. These can be converted to 

 samples of the uncompressed speech by complementary instantaneous ex- 

 pansion. The spectrum of these samples is shown in Fig. 2(c). All that is 

 necessary at this point to recover the original speech without distortion is 

 to pass these samples through a 4 kc. low-pass filter. 



Requirements for Distortionless Transmission on Reduced 

 Bandwidth Basis 



Thus the criterion for distortionless transmission of compressed and 

 sampled speech under these conditions is that the samples recovered at the 

 receiver be the same as the samples of compressed speech that were generated 

 at the transmitter. This means sending 8000 pulses per second over a 4 kc 

 band without intersymbol interference. Nyquist' has shown that this is the 

 maximum rate at which independent pulses can be transmitted over a 4 

 kc band and still be recovered at the receiver. At this maximum rate, the 

 bandwidth employed does not give the transient response of one pulse time 

 to die out before the next pulse is received. Therefore the transient response 

 in this case must be such that, when one pulse is at its peak, the transient 

 responses of all other pulses will be going through zero. The infinitely sharp 

 cut-ofif at 4 kc, which is required to separate out the spectrum shown in 

 Fig. 2(a) from that in Fig. 1(d), will have the required zeros in its pulse 

 response, provided the attenuation is constant and the phase is linear with 

 frequency. 



This is the familiar shape of transient response. Nyquist has shown 



x 



3H. Nyquist, "Certain Topics in Telegraph Transmission Theory," A.LE.E. Trans- 

 actions, Vol. 47, Pages 617 to 644, April 1928. 



