By N. Bourbaki

Ce dixiÃ¨me chapitre du Livre d AlgÃ¨bre, deuxiÃ¨me Livre du traitÃ©, pose les bases du calcul homologique.

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Extra info for Algebre: Chapitre 10.Algebre homologique

Sample text

15) that is, the real and imaginary parts of eiwon vary sinusoidally with n. By analogy with the continuous-time case, the quantity w 0 is called the frequency of the complex sinusoid or complex exponential, and 4> is called the phase. However, note that n is a dimensionless integer. Thus, the dimension of w 0 must be radians. If we wish to maintain a closer analogy with the continuous-time case, we can specify the units of w 0 to be radians per sample and the units of n to be samples. The fact that n is always an integer inEq.

Also, it follows from Eq. each cascade system is the convolution of the individual impulse responses. 79) = o[n]- 8[n -1]. 13(c). 13(a) and (b) have been converted to causal systems by cascading them with a delay. In general, any noncausal FIR system can be made causal by cascading it with a sufficiently long delay. Another example of cascaded systems introduces the concept of an inverse system. 14. The impulse response of the cascade system is h[n] = u[n] = * (8[n]- 8[n -1]) u[n J - u[n - 1] f'1 QO\ \"-'•UVJ = 8[n].

Ibis then provides sufficient initial conditions to obtain y[n] for n :::: n0 recursively using Eq. 97). To summarize, for a system for which the input and output satisfy a linear constantcoefficient difference equation: • The output for a given input is not uniquely specified. Auxiliary information or conditions are required. • If the auxiliary information is in the form of N sequential values of the output, later values can be obtained by rearranging the difference equation as a recursive relation running forward in n, and prior values can be obtained by rearranging the difference equation as a recursive relation running backward in n.