6. Frequency-Domain Analysis of Discrete-Time Signals and Systems.pdf

6. Frequency-Domain Analysis of Discrete- Time Signals and Systems 6.1. Properties of Sequence exp(jn) (1.3.3) 6.2. Definition of Discrete-Time Fourier Series (3.6) 6.3. Properties of Discrete-Time Fourier Series (3.7) 6.4. Definition of Discrete-Time Fourier Transform (5.0-5.2) 6.5. Properties of Discrete-Time Fourier Transform (5.3-5.7) 6.6. Frequency Response (3.2, 3.8, 5.4) 6.7. Linear Constant-Coefficient Difference Equations (5.8) 6.1. Properties of Sequence exp(jn) 6.1.1. Periodicity of Sequence exp(jn) The sequence exp(j n) is periodic if and only if can be written as k  2 , (6.1) N where k and N are integers. It can be shown that N is a period of the sequence. If N0, and k and N have no factors in common, N will be the fundamental period of the sequence. Note that exp(jt) is always periodic. Example. Determine the periodicity of the following signals: (1) x(t)=cos(t). (2) x(t)=exp(jt). (3) x(n)=cos(n). Example. Find the fundamental period of the following sequence:  2   3  x(n) exp j n exp j n . (6.2)      3   4  6.1.2. Frequency of Sequence exp(jn) is referred to as the frequency of sequence exp(jn). It is equal to the variation of the phase n in a sampling interval, and describes how fast the phase n changes. exp(jn)=exp(jn) if and only if =a multiple of 2. Note 1 2 1 2 that exp(jt)=exp(jt) if and only if = . 1 2 1 2 Example. Several pairs of signals are given next. Determine if the two signals in each pair are identical