15.1 Causal Systems 369 LTI-systems w e t irne-invariant as weU as being linear, a i d so thr mnditionr (15.3) and (15.4) only need to l-te fornmlated for one point in time for cxampk, fo = or ko = 0, obtairiirig for corltimlous and discrete systems: it) = tor f < r [ k ]= (I for k < * g(t) = ===+ y[k] = for k for t < i (15.5) (15.6) With help from t h e coilvolution relationships (8.39) arid (14.33), y(t) = h(t)* s ( f ) - h( z ) x ( t - ) d r (15.7) rx 33 g[k] = h [ k ]* r [ k ] = k [ K ] X [ k- K ] (15.8) h=-W we can cxpress the coaditions (15.5) arid (15.6) significantly more concisely using the iniptilse respoiase For ront,ixiuuus sy ins it follows from (15.7) with z ( t ) = for t < that (1 5.9) 'l'bc r n u s a l i t y condition y ( t ) - for t < is then fulfilled when tlie impulse resporrse h(t) disappears for t < 0: h ( / )= for t