数字信号处理Chapter 6.ppt

Convergence condition of DTFT x[n] is an absolutely summable sequence mean-square convergence Dirac delta function δ(ω) z-Transform The DTFT provides a frequency-domain representation of discrete-time signals and LTI discrete-time systems Because of the convergence condition, in many cases, the DTFT of a sequence may not exist As a result, it is not possible to make use of such frequency-domain characterization in these cases z-Transform A generalization of the DTFT defined by leads to the z-transform z-transform may exist for many sequences for which the DTFT does not exist Moreover, use of z-transform techniques permits simple algebraic manipulations Definition and Properties Consequently, z-transform has become an important tool in the analysis and design of digital filters For a given sequence g[n], its z-transform G(z) is defined as where is a complex variable Definition and Properties If we let , then the z-transform reduces to The above can be interpreted as the DTFT of the modified sequence For r = 1 (i.e., |z| = 1), z-transform reduces to its DTFT, provided the latter exists Definition and Properties The contour |z| = 1 is a circle in the z-plane of unity radius and is called the unit circle Like the DTFT, there are conditions on the convergence of the infinite series For a given sequence, the set R of values of z for which its z-transform converges is called the region of convergence (ROC) Definition and Properties From our earlier discussion on the uniform convergence of the DTFT, it follows that the series converges if is absolutely summable, i.e., if Definition and Properties In general, the ROC of a z-transform of a sequence g[n] is an annular annular region of the z-plane: where Definition and Properties Example - Determine the z-transform X(z) of the causal sequence and its ROC Now The above power series converges to ROC is the annul