Note: This blog is based on the second edition of Oppenheim's "Signals and Systems", mainly for the review and deepening of their own learning.
1. Representation of aperiodic signals: discrete-time Borier transform
1), the derivation of the discrete-time Borier transform
1. Discrete-time Borier transform pair
1), X(ejw) is called the discrete-time Borier transform, and this pair of expressions is the discrete-time Borier transform pair.
2) The above formula is called a comprehensive formula, and the following formula is called an analysis formula.
2. The discrete-time Borrie transform has many similarities with the continuous-time case. The main difference between the two is the periodicity of the discrete-time transform X(ejw) and the finite integration interval in the synthesis formula.
2), on the convergence problem of discrete-time Borier transform
1. When the signal is not on-site, the following formula must be considered
Convergence of the sum of infinite terms in . If x[n] is absolutely summable, i.e.
Or, if the energy of this sequence is finite, i.e.
Then it must converge.
2. The following formula
The integration of is carried out in a finite integration interval, so there is no convergence problem.
2. The Borier transform of the periodic washing of your number
1. Consider the following signals
The Bolier transform of x[n] is formally the following impulse train
2. Consider a periodic sequence x[n], the period is N, and its Borier series is
At this time, the Borrie transform is
In this way, the Bolier transform of a periodic signal can be obtained directly from its Bolier coefficients.
3. The system represented by the linear constant coefficient differential method
1. For a fragrance time-invariant system, the linear constant coefficient difference equation between the output y[n] and the input x[n] generally has the following implementation:
, the difference equation of this formula is generally called the Nth order difference equation. There are two ways to determine H(ejw). The first of these is obtained by exploiting the fact that the complex exponent is an eigenfunction of a linear time-invariant system. The second is to use the convolution, linearity and time-shift properties of the discrete-time Borier transform.
2. Let X (ejw ), Y (ejw ) and H (ejw ) be the Borier transform of output x[n], output y[n] and unit impulse response h[n] respectively, then the discrete-time Borier transform The convolutional nature of the leaf transform means that there are
