Discrete-time signals and systems: 4

12. System

A system is a transformation (transformations, a description of motion expressed in linear algebra)

A discrete time domain signal x is mapped to a discrete time domain signal y. For example: magnetic resonance imaging System.

Where is offset translated into: compensation system? Decimate translation into a sampling system?

Summarize:

The system transforms one signal into another (transform) signal by manipulating the information.

Only consider:

Transforms an infinite signal x into an infinite signal y.

The signal x with a length of n is transformed into a signal y of length n.

12. Linear system

Two properties: Scalability and additive.

Note: The two properties are the system output y changes with the system input x. The overall change of x[n] in the corresponding signal is not the change of N. (Time invariant is about N)

To prove the linearity of the system, it is necessary to prove that the system maintains two properties for any input signal. And proving nonlinearity requires only one return example.

Matrix multiplication and Linear systems

Written in front: linear algebra is the discipline that describes linear change, the main means is matrix multiplication. So they contacted nature closely.

Matrix multiplication is a linear system.

All linear systems can be expressed as matrix multiplication.  This conclusion is very imba, think of engineering research, are all kinds of model systems, and these systems are basically linear system!!! So, all matrices are multiplied.

It is interesting to express the linear system in the form of an image. And the expression of image is one of the methods of many linear systems.

The linear system can be regarded as the linear combination of the matrix H's column vectors, x is the weight vector, and y is the output.

Linear systems can also be seen as a series of inner product. Each y[n] is the inner product of the nth row and x in the H matrix.

In addition to both of these understandings:

From the point of view of linear algebra, the multiplication of matrices is the linear transformation of a vector (from one point of space to another), or the point is invariant, but the transformation of the space itself. RELATED Links: Matrix understanding of this article.

Summarize:

The nature of the linear system.

The linear system H can be expressed as a matrix (h), so it can be understood from two angles:

The system output Y can be regarded as the weighted average of the matrix H, and the weight is x.

The system output Y can also be seen as a sequence of H's line vectors and the inner product of X.

Time-invariant system (also called shift-invariant system)

A system input x yesterday, today input x, tomorrow input x, any time input x will only get the same output Y.

The output is only related to X, not time-dependent, why emphasize this? Because the signal is x[n] is the function of time. so the time-invariant system has nothing to do with the change of n!!!

The same time displacement produces the same time displacement output. The same time displacement produces the same time displacement output. The same time displacement produces the same time displacement output.

Time-invariant systems (expression of finite signals)

The difference from the infinite signal is that the shift of the finite signal is done by means of a die Operation (ring) , remember? (in the first chapter). Therefore, the modulo operation is used on an expression.

Summarize:

The nature of the time-invariant system is constant, no matter what time it is entered.

Infinite signal: The time shift of any integer, the system remains unchanged.

Finite signal: Any annular time displacement, the system remains unchanged.

Linear is the invariant system

It also contains linear and time-invariant systems of two properties.

Interesting place to appear!!! Interesting place to appear!!! Interesting place to appear!!!

The matrix H of linear time-invariant systems is very special.

Output Y[N] is the nth line of H, multiplied by the inner product of the X-signal, i.e. each hn,m and x[m]. (because the output of the system is only related to the input x, and H is unchanged, that is, H does not change)

When the system is a time-invariant system, the N-Q is brought into the system, resulting in great changes.

The matrix H consisting of H is called the gnas matrix gnas matrix gnas matrix .

The characteristic of the matrix is that the elements are all the same in the diagonal direction

This nature allows the information of the entire matrix to be stored through a single vector.

The No. 0 column contains all the information (note the position of 0, at the center of the Matrix), and the other elements of the matrix are copied diagonally.

In addition, the time is reversed in line No. 0 (line No. 0, the time of the row is reversed).

Based on the properties above, a new matrix representation is obtained, the value of each element Hn,m=h[n-m]

Matrix structure (finite-length signal) of the line row invariant system-matrix Structure of LTI Systems (Finite-length signals)

An input signal of length n, through a linear system, obtains an output signal of length n. The matrix is expressed as: the nth output signal y[n] equals the nth row of the H matrix, and each corresponding m element of the line is multiplied by the first element of the input signal X, and then added.

For the system to join the invariant nature, that is, input x[n-q] corresponding output Y[N-Q]. So the system H matrix does not change, and the original M element becomes m-q ( note the conversion of M and N, because H uses N to represent the row, so the number of elements is represented by M ), by means of modulo operations: (M-Q) n

A further substitution of n ' =n-q m ' =m-q is to replace the expression of the system with the form of x[n] and Y[n], paying attention to the change of H coordinate.

So, we get the input as x[n], the corresponding output is y[n], so the description H matrix has a special form as follows:

All bands of the Matrix are shown in the same diagonal direction:

Because the elements in the matrix are the result of constant ring-mode operations, the elements of column o contain information about the entire matrix (corresponding to the circular sequence reversal of the elements of line No. 0).

Note: The following matrix only needs an ordinal n, which is too useful in the calculation of convolution!!! This is too useful in the calculation of convolution!! This is too useful in the calculation of convolution!!

An image representation of the above results:

Summarize:

Linear time-invariant system = linear + time invariant. Is the basis of the signal processing system.

Infinite signaling system = Gnas matrix (Toeplitz matrix H), the matrix has a special form:

Correspondingly, the system of the finite signal is the cyclic matrix (circulant matrix)

It is important that these two matrices no longer use subscripts to represent matrices.

Discrete-time signals and systems: 4