package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Check state-space matrices and ensure they are two-dimensional.

If enough information on the system is provided, that is, enough properly-shaped arrays are passed to the function, the missing ones are built from this information, ensuring the correct number of rows and columns. Otherwise a ValueError is raised.

Parameters ---------- A, B, C, D : array_like, optional State-space matrices. All of them are None (missing) by default. See `ss2tf` for format.

Returns ------- A, B, C, D : array Properly shaped state-space matrices.

Raises ------ ValueError If not enough information on the system was provided.

array(object, dtype=None, copy=True, order='K', subok=False, ndmin=0)

Create an array.

Parameters ---------- object : array_like An array, any object exposing the array interface, an object whose __array__ method returns an array, or any (nested) sequence. dtype : data-type, optional The desired data-type for the array. If not given, then the type will be determined as the minimum type required to hold the objects in the sequence. copy : bool, optional If true (default), then the object is copied. Otherwise, a copy will only be made if __array__ returns a copy, if obj is a nested sequence, or if a copy is needed to satisfy any of the other requirements (`dtype`, `order`, etc.). order : 'K', 'A', 'C', 'F', optional Specify the memory layout of the array. If object is not an array, the newly created array will be in C order (row major) unless 'F' is specified, in which case it will be in Fortran order (column major). If object is an array the following holds.

===== ========= =================================================== order no copy copy=True ===== ========= =================================================== 'K' unchanged F & C order preserved, otherwise most similar order 'A' unchanged F order if input is F and not C, otherwise C order 'C' C order C order 'F' F order F order ===== ========= ===================================================

When ``copy=False`` and a copy is made for other reasons, the result is the same as if ``copy=True``, with some exceptions for `A`, see the Notes section. The default order is 'K'. subok : bool, optional If True, then sub-classes will be passed-through, otherwise the returned array will be forced to be a base-class array (default). ndmin : int, optional Specifies the minimum number of dimensions that the resulting array should have. Ones will be pre-pended to the shape as needed to meet this requirement.

Returns ------- out : ndarray An array object satisfying the specified requirements.

See Also -------- empty_like : Return an empty array with shape and type of input. ones_like : Return an array of ones with shape and type of input. zeros_like : Return an array of zeros with shape and type of input. full_like : Return a new array with shape of input filled with value. empty : Return a new uninitialized array. ones : Return a new array setting values to one. zeros : Return a new array setting values to zero. full : Return a new array of given shape filled with value.

Notes ----- When order is 'A' and `object` is an array in neither 'C' nor 'F' order, and a copy is forced by a change in dtype, then the order of the result is not necessarily 'C' as expected. This is likely a bug.

Examples -------- >>> np.array(1, 2, 3) array(1, 2, 3)

Upcasting:

>>> np.array(1, 2, 3.0) array( 1., 2., 3.)

More than one dimension:

>>> np.array([1, 2], [3, 4]) array([1, 2], [3, 4])

Minimum dimensions 2:

>>> np.array(1, 2, 3, ndmin=2) array([1, 2, 3])

Type provided:

>>> np.array(1, 2, 3, dtype=complex) array( 1.+0.j, 2.+0.j, 3.+0.j)

Data-type consisting of more than one element:

>>> x = np.array((1,2),(3,4),dtype=('a','<i4'),('b','<i4')) >>> x'a' array(1, 3)

Creating an array from sub-classes:

>>> np.array(np.mat('1 2; 3 4')) array([1, 2], [3, 4])

>>> np.array(np.mat('1 2; 3 4'), subok=True) matrix([1, 2], [3, 4])

Convert the input to an array.

Parameters ---------- a : array_like Input data, in any form that can be converted to an array. This includes lists, lists of tuples, tuples, tuples of tuples, tuples of lists and ndarrays. dtype : data-type, optional By default, the data-type is inferred from the input data. order : 'C', 'F', optional Whether to use row-major (C-style) or column-major (Fortran-style) memory representation. Defaults to 'C'.

Returns ------- out : ndarray Array interpretation of `a`. No copy is performed if the input is already an ndarray with matching dtype and order. If `a` is a subclass of ndarray, a base class ndarray is returned.

See Also -------- asanyarray : Similar function which passes through subclasses. ascontiguousarray : Convert input to a contiguous array. asfarray : Convert input to a floating point ndarray. asfortranarray : Convert input to an ndarray with column-major memory order. asarray_chkfinite : Similar function which checks input for NaNs and Infs. fromiter : Create an array from an iterator. fromfunction : Construct an array by executing a function on grid positions.

Examples -------- Convert a list into an array:

>>> a = 1, 2 >>> np.asarray(a) array(1, 2)

Existing arrays are not copied:

>>> a = np.array(1, 2) >>> np.asarray(a) is a True

If `dtype` is set, array is copied only if dtype does not match:

>>> a = np.array(1, 2, dtype=np.float32) >>> np.asarray(a, dtype=np.float32) is a True >>> np.asarray(a, dtype=np.float64) is a False

Contrary to `asanyarray`, ndarray subclasses are not passed through:

>>> issubclass(np.recarray, np.ndarray) True >>> a = np.array((1.0, 2), (3.0, 4), dtype='f4,i4').view(np.recarray) >>> np.asarray(a) is a False >>> np.asanyarray(a) is a True

View inputs as arrays with at least two dimensions.

Parameters ---------- arys1, arys2, ... : array_like One or more array-like sequences. Non-array inputs are converted to arrays. Arrays that already have two or more dimensions are preserved.

Returns ------- res, res2, ... : ndarray An array, or list of arrays, each with ``a.ndim >= 2``. Copies are avoided where possible, and views with two or more dimensions are returned.

See Also -------- atleast_1d, atleast_3d

Examples -------- >>> np.atleast_2d(3.0) array([3.])

>>> x = np.arange(3.0) >>> np.atleast_2d(x) array([0., 1., 2.]) >>> np.atleast_2d(x).base is x True

>>> np.atleast_2d(1, 1, 2, [1, 2]) array([[1]]), array([[1, 2]]), array([[1, 2]])

Transform a continuous to a discrete state-space system.

Parameters ---------- system : a tuple describing the system or an instance of `lti` The following gives the number of elements in the tuple and the interpretation:

* 1: (instance of `lti`) * 2: (num, den) * 3: (zeros, poles, gain) * 4: (A, B, C, D)

dt : float The discretization time step. method : str, optional Which method to use:

* gbt: generalized bilinear transformation * bilinear: Tustin's approximation ('gbt' with alpha=0.5) * euler: Euler (or forward differencing) method ('gbt' with alpha=0) * backward_diff: Backwards differencing ('gbt' with alpha=1.0) * zoh: zero-order hold (default) * foh: first-order hold ( *versionadded: 1.3.0* ) * impulse: equivalent impulse response ( *versionadded: 1.3.0* )

alpha : float within 0, 1, optional The generalized bilinear transformation weighting parameter, which should only be specified with method='gbt', and is ignored otherwise

Returns ------- sysd : tuple containing the discrete system Based on the input type, the output will be of the form

* (num, den, dt) for transfer function input * (zeros, poles, gain, dt) for zeros-poles-gain input * (A, B, C, D, dt) for state-space system input

Notes ----- By default, the routine uses a Zero-Order Hold (zoh) method to perform the transformation. Alternatively, a generalized bilinear transformation may be used, which includes the common Tustin's bilinear approximation, an Euler's method technique, or a backwards differencing technique.

The Zero-Order Hold (zoh) method is based on 1_, the generalized bilinear approximation is based on 2_ and 3_, the First-Order Hold (foh) method is based on 4_.

References ---------- .. 1 https://en.wikipedia.org/wiki/Discretization#Discretization_of_linear_state_space_models

.. 2 http://techteach.no/publications/discretetime_signals_systems/discrete.pdf

.. 3 G. Zhang, X. Chen, and T. Chen, Digital redesign via the generalized bilinear transformation, Int. J. Control, vol. 82, no. 4, pp. 741-754, 2009. (https://www.mypolyuweb.hk/~magzhang/Research/ZCC09_IJC.pdf)

.. 4 G. F. Franklin, J. D. Powell, and M. L. Workman, Digital control of dynamic systems, 3rd ed. Menlo Park, Calif: Addison-Wesley, pp. 204-206, 1998.

dot(a, b, out=None)

Dot product of two arrays. Specifically,

• If both `a` and `b` are 1-D arrays, it is inner product of vectors (without complex conjugation).

• If both `a` and `b` are 2-D arrays, it is matrix multiplication, but using :func:`matmul` or ``a @ b`` is preferred.

• If either `a` or `b` is 0-D (scalar), it is equivalent to :func:`multiply` and using ``numpy.multiply(a, b)`` or ``a * b`` is preferred.

• If `a` is an N-D array and `b` is a 1-D array, it is a sum product over the last axis of `a` and `b`.

• If `a` is an N-D array and `b` is an M-D array (where ``M>=2``), it is a sum product over the last axis of `a` and the second-to-last axis of `b`::

dot(a, b)i,j,k,m = sum(ai,j,: * bk,:,m)

Parameters ---------- a : array_like First argument. b : array_like Second argument. out : ndarray, optional Output argument. This must have the exact kind that would be returned if it was not used. In particular, it must have the right type, must be C-contiguous, and its dtype must be the dtype that would be returned for `dot(a,b)`. This is a performance feature. Therefore, if these conditions are not met, an exception is raised, instead of attempting to be flexible.

Returns ------- output : ndarray Returns the dot product of `a` and `b`. If `a` and `b` are both scalars or both 1-D arrays then a scalar is returned; otherwise an array is returned. If `out` is given, then it is returned.

Raises ------ ValueError If the last dimension of `a` is not the same size as the second-to-last dimension of `b`.

See Also -------- vdot : Complex-conjugating dot product. tensordot : Sum products over arbitrary axes. einsum : Einstein summation convention. matmul : '@' operator as method with out parameter.

Examples -------- >>> np.dot(3, 4) 12

Neither argument is complex-conjugated:

>>> np.dot(2j, 3j, 2j, 3j) (-13+0j)

For 2-D arrays it is the matrix product:

>>> a = [1, 0], [0, 1] >>> b = [4, 1], [2, 2] >>> np.dot(a, b) array([4, 1], [2, 2])

>>> a = np.arange(3*4*5*6).reshape((3,4,5,6)) >>> b = np.arange(3*4*5*6)::-1.reshape((5,4,6,3)) >>> np.dot(a, b)2,3,2,1,2,2 499128 >>> sum(a2,3,2,: * b1,2,:,2) 499128

Return a 2-D array with ones on the diagonal and zeros elsewhere.

Parameters ---------- N : int Number of rows in the output. M : int, optional Number of columns in the output. If None, defaults to `N`. k : int, optional Index of the diagonal: 0 (the default) refers to the main diagonal, a positive value refers to an upper diagonal, and a negative value to a lower diagonal. dtype : data-type, optional Data-type of the returned array. order : 'C', 'F', optional Whether the output should be stored in row-major (C-style) or column-major (Fortran-style) order in memory.

.. versionadded:: 1.14.0

Returns ------- I : ndarray of shape (N,M) An array where all elements are equal to zero, except for the `k`-th diagonal, whose values are equal to one.

See Also -------- identity : (almost) equivalent function diag : diagonal 2-D array from a 1-D array specified by the user.

Examples -------- >>> np.eye(2, dtype=int) array([1, 0], [0, 1]) >>> np.eye(3, k=1) array([0., 1., 0.], [0., 0., 1.], [0., 0., 0.])

Normalize numerator/denominator of a continuous-time transfer function.

If values of `b` are too close to 0, they are removed. In that case, a BadCoefficients warning is emitted.

Parameters ---------- b: array_like Numerator of the transfer function. Can be a 2d array to normalize multiple transfer functions. a: array_like Denominator of the transfer function. At most 1d.

Returns ------- num: array The numerator of the normalized transfer function. At least a 1d array. A 2d-array if the input `num` is a 2d array. den: 1d-array The denominator of the normalized transfer function.

Notes ----- Coefficients for both the numerator and denominator should be specified in descending exponent order (e.g., ``s^2 + 3s + 5`` would be represented as ``1, 3, 5``).

Compute the outer product of two vectors.

Given two vectors, ``a = a0, a1, ..., aM`` and ``b = b0, b1, ..., bN``, the outer product 1_ is::

[a0*b0 a0*b1 ... a0*bN ] [a1*b0 . [ ... . [aM*b0 aM*bN ]] Parameters ---------- a : (M,) array_like First input vector. Input is flattened if not already 1-dimensional. b : (N,) array_like Second input vector. Input is flattened if not already 1-dimensional. out : (M, N) ndarray, optional A location where the result is stored .. versionadded:: 1.9.0 Returns ------- out : (M, N) ndarray ``out[i, j] = a[i] * b[j]`` See also -------- inner einsum : ``einsum('i,j->ij', a.ravel(), b.ravel())`` is the equivalent. ufunc.outer : A generalization to N dimensions and other operations. ``np.multiply.outer(a.ravel(), b.ravel())`` is the equivalent. References ---------- .. [1] : G. H. Golub and C. F. Van Loan, *Matrix Computations*, 3rd ed., Baltimore, MD, Johns Hopkins University Press, 1996, pg. 8. Examples -------- Make a ( *very* coarse) grid for computing a Mandelbrot set: >>> rl = np.outer(np.ones((5,)), np.linspace(-2, 2, 5)) >>> rl array([[-2., -1., 0., 1., 2.], [-2., -1., 0., 1., 2.], [-2., -1., 0., 1., 2.], [-2., -1., 0., 1., 2.], [-2., -1., 0., 1., 2.]]) >>> im = np.outer(1j*np.linspace(2, -2, 5), np.ones((5,))) >>> im array([[0.+2.j, 0.+2.j, 0.+2.j, 0.+2.j, 0.+2.j], [0.+1.j, 0.+1.j, 0.+1.j, 0.+1.j, 0.+1.j], [0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], [0.-1.j, 0.-1.j, 0.-1.j, 0.-1.j, 0.-1.j], [0.-2.j, 0.-2.j, 0.-2.j, 0.-2.j, 0.-2.j]]) >>> grid = rl + im >>> grid array([[-2.+2.j, -1.+2.j, 0.+2.j, 1.+2.j, 2.+2.j], [-2.+1.j, -1.+1.j, 0.+1.j, 1.+1.j, 2.+1.j], [-2.+0.j, -1.+0.j, 0.+0.j, 1.+0.j, 2.+0.j], [-2.-1.j, -1.-1.j, 0.-1.j, 1.-1.j, 2.-1.j], [-2.-2.j, -1.-2.j, 0.-2.j, 1.-2.j, 2.-2.j]]) An example using a 'vector' of letters: >>> x = np.array(['a', 'b', 'c'], dtype=object) >>> np.outer(x, [1, 2, 3]) array([['a', 'aa', 'aaa'], ['b', 'bb', 'bbb'], ['c', 'cc', 'ccc']], dtype=object)

Find the coefficients of a polynomial with the given sequence of roots.

Returns the coefficients of the polynomial whose leading coefficient is one for the given sequence of zeros (multiple roots must be included in the sequence as many times as their multiplicity; see Examples). A square matrix (or array, which will be treated as a matrix) can also be given, in which case the coefficients of the characteristic polynomial of the matrix are returned.

Parameters ---------- seq_of_zeros : array_like, shape (N,) or (N, N) A sequence of polynomial roots, or a square array or matrix object.

Returns ------- c : ndarray 1D array of polynomial coefficients from highest to lowest degree:

``c0 * x**(N) + c1 * x**(N-1) + ... + cN-1 * x + cN`` where c0 always equals 1.

Raises ------ ValueError If input is the wrong shape (the input must be a 1-D or square 2-D array).

See Also -------- polyval : Compute polynomial values. roots : Return the roots of a polynomial. polyfit : Least squares polynomial fit. poly1d : A one-dimensional polynomial class.

Notes ----- Specifying the roots of a polynomial still leaves one degree of freedom, typically represented by an undetermined leading coefficient. 1_ In the case of this function, that coefficient - the first one in the returned array - is always taken as one. (If for some reason you have one other point, the only automatic way presently to leverage that information is to use ``polyfit``.)

The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n` matrix **A** is given by

:math:`p_a(t) = \mathrmdet(t\, \mathbfI - \mathbfA)`,

where **I** is the `n`-by-`n` identity matrix. 2_

References ---------- .. 1 M. Sullivan and M. Sullivan, III, 'Algebra and Trignometry, Enhanced With Graphing Utilities,' Prentice-Hall, pg. 318, 1996.

.. 2 G. Strang, 'Linear Algebra and Its Applications, 2nd Edition,' Academic Press, pg. 182, 1980.

Examples -------- Given a sequence of a polynomial's zeros:

>>> np.poly((0, 0, 0)) # Multiple root example array(1., 0., 0., 0.)

The line above represents z**3 + 0*z**2 + 0*z + 0.

>>> np.poly((-1./2, 0, 1./2)) array( 1. , 0. , -0.25, 0. )

The line above represents z**3 - z/4

>>> np.poly((np.random.random(1)0, 0, np.random.random(1)0)) array( 1. , -0.77086955, 0.08618131, 0. ) # random

Given a square array object:

>>> P = np.array([0, 1./3], [-1./2, 0]) >>> np.poly(P) array(1. , 0. , 0.16666667)

Note how in all cases the leading coefficient is always 1.

Return the product of array elements over a given axis.

Parameters ---------- a : array_like Input data. axis : None or int or tuple of ints, optional Axis or axes along which a product is performed. The default, axis=None, will calculate the product of all the elements in the input array. If axis is negative it counts from the last to the first axis.

.. versionadded:: 1.7.0

If axis is a tuple of ints, a product is performed on all of the axes specified in the tuple instead of a single axis or all the axes as before. dtype : dtype, optional The type of the returned array, as well as of the accumulator in which the elements are multiplied. The dtype of `a` is used by default unless `a` has an integer dtype of less precision than the default platform integer. In that case, if `a` is signed then the platform integer is used while if `a` is unsigned then an unsigned integer of the same precision as the platform integer is used. out : ndarray, optional Alternative output array in which to place the result. It must have the same shape as the expected output, but the type of the output values will be cast if necessary. keepdims : bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

If the default value is passed, then `keepdims` will not be passed through to the `prod` method of sub-classes of `ndarray`, however any non-default value will be. If the sub-class' method does not implement `keepdims` any exceptions will be raised. initial : scalar, optional The starting value for this product. See `~numpy.ufunc.reduce` for details.

.. versionadded:: 1.15.0

where : array_like of bool, optional Elements to include in the product. See `~numpy.ufunc.reduce` for details.

.. versionadded:: 1.17.0

Returns ------- product_along_axis : ndarray, see `dtype` parameter above. An array shaped as `a` but with the specified axis removed. Returns a reference to `out` if specified.

See Also -------- ndarray.prod : equivalent method ufuncs-output-type

Notes ----- Arithmetic is modular when using integer types, and no error is raised on overflow. That means that, on a 32-bit platform:

>>> x = np.array(536870910, 536870910, 536870910, 536870910) >>> np.prod(x) 16 # may vary

The product of an empty array is the neutral element 1:

>>> np.prod() 1.0

Examples -------- By default, calculate the product of all elements:

>>> np.prod(1.,2.) 2.0

Even when the input array is two-dimensional:

>>> np.prod([1.,2.],[3.,4.]) 24.0

But we can also specify the axis over which to multiply:

>>> np.prod([1.,2.],[3.,4.], axis=1) array( 2., 12.)

Or select specific elements to include:

>>> np.prod(1., np.nan, 3., where=True, False, True) 3.0

If the type of `x` is unsigned, then the output type is the unsigned platform integer:

>>> x = np.array(1, 2, 3, dtype=np.uint8) >>> np.prod(x).dtype == np.uint True

If `x` is of a signed integer type, then the output type is the default platform integer:

>>> x = np.array(1, 2, 3, dtype=np.int8) >>> np.prod(x).dtype == int True

You can also start the product with a value other than one:

>>> np.prod(1, 2, initial=5) 10

State-space to transfer function.

A, B, C, D defines a linear state-space system with `p` inputs, `q` outputs, and `n` state variables.

Parameters ---------- A : array_like State (or system) matrix of shape ``(n, n)`` B : array_like Input matrix of shape ``(n, p)`` C : array_like Output matrix of shape ``(q, n)`` D : array_like Feedthrough (or feedforward) matrix of shape ``(q, p)`` input : int, optional For multiple-input systems, the index of the input to use.

Returns ------- num : 2-D ndarray Numerator(s) of the resulting transfer function(s). `num` has one row for each of the system's outputs. Each row is a sequence representation of the numerator polynomial. den : 1-D ndarray Denominator of the resulting transfer function(s). `den` is a sequence representation of the denominator polynomial.

Examples -------- Convert the state-space representation:

.. math::

\dot\textbf{x

}

(t) = \beginmatrix -2 & -1 \\ 1 & 0 \endmatrix \textbfx(t) + \beginmatrix 1 \\ 0 \endmatrix \textbfu(t) \\

\textbfy(t) = \beginmatrix 1 & 2 \endmatrix \textbfx(t) + \beginmatrix 1 \endmatrix \textbfu(t)

>>> A = [-2, -1], [1, 0] >>> B = [1], [0] # 2-dimensional column vector >>> C = [1, 2] # 2-dimensional row vector >>> D = 1

to the transfer function:

.. math:: H(s) = \fracs^2 + 3s + 3s^2 + 2s + 1

>>> from scipy.signal import ss2tf >>> ss2tf(A, B, C, D) (array([1, 3, 3]), array( 1., 2., 1.))

State-space representation to zero-pole-gain representation.

A, B, C, D defines a linear state-space system with `p` inputs, `q` outputs, and `n` state variables.

Parameters ---------- A : array_like State (or system) matrix of shape ``(n, n)`` B : array_like Input matrix of shape ``(n, p)`` C : array_like Output matrix of shape ``(q, n)`` D : array_like Feedthrough (or feedforward) matrix of shape ``(q, p)`` input : int, optional For multiple-input systems, the index of the input to use.

Returns ------- z, p : sequence Zeros and poles. k : float System gain.

Transfer function to state-space representation.

Parameters ---------- num, den : array_like Sequences representing the coefficients of the numerator and denominator polynomials, in order of descending degree. The denominator needs to be at least as long as the numerator.

Returns ------- A, B, C, D : ndarray State space representation of the system, in controller canonical form.

Examples -------- Convert the transfer function:

.. math:: H(s) = \fracs^2 + 3s + 3s^2 + 2s + 1

>>> num = 1, 3, 3 >>> den = 1, 2, 1

to the state-space representation:

.. math::

\dot\textbf{x

}

(t) = \beginmatrix -2 & -1 \\ 1 & 0 \endmatrix \textbfx(t) + \beginmatrix 1 \\ 0 \endmatrix \textbfu(t) \\

\textbfy(t) = \beginmatrix 1 & 2 \endmatrix \textbfx(t) + \beginmatrix 1 \endmatrix \textbfu(t)

>>> from scipy.signal import tf2ss >>> A, B, C, D = tf2ss(num, den) >>> A array([-2., -1.], [ 1., 0.]) >>> B array([ 1.], [ 0.]) >>> C array([ 1., 2.]) >>> D array([ 1.])

Return zero, pole, gain (z, p, k) representation from a numerator, denominator representation of a linear filter.

Parameters ---------- b : array_like Numerator polynomial coefficients. a : array_like Denominator polynomial coefficients.

Returns ------- z : ndarray Zeros of the transfer function. p : ndarray Poles of the transfer function. k : float System gain.

Notes ----- If some values of `b` are too close to 0, they are removed. In that case, a BadCoefficients warning is emitted.

The `b` and `a` arrays are interpreted as coefficients for positive, descending powers of the transfer function variable. So the inputs :math:`b = b_0, b_1, ..., b_M` and :math:`a =a_0, a_1, ..., a_N` can represent an analog filter of the form:

.. math::

H(s) = \frac _0 s^M + b_1 s^(M-1) + \cdots + b_M a_0 s^N + a_1 s^{(N-1) + \cdots + a_N

}

or a discrete-time filter of the form:

.. math::

H(z) = \frac _0 z^M + b_1 z^(M-1) + \cdots + b_M a_0 z^N + a_1 z^{(N-1) + \cdots + a_N

}

This 'positive powers' form is found more commonly in controls engineering. If `M` and `N` are equal (which is true for all filters generated by the bilinear transform), then this happens to be equivalent to the 'negative powers' discrete-time form preferred in DSP:

.. math::

H(z) = \frac _0 + b_1 z^

1

}

1. \cdots + b_M z^

   M

}

}

a_0 + a_1 z^{-1 + \cdots + a_N z^

N

}

}

Although this is true for common filters, remember that this is not true in the general case. If `M` and `N` are not equal, the discrete-time transfer function coefficients must first be converted to the 'positive powers' form before finding the poles and zeros.

zeros(shape, dtype=float, order='C')

Return a new array of given shape and type, filled with zeros.

Parameters ---------- shape : int or tuple of ints Shape of the new array, e.g., ``(2, 3)`` or ``2``. dtype : data-type, optional The desired data-type for the array, e.g., `numpy.int8`. Default is `numpy.float64`. order : 'C', 'F', optional, default: 'C' Whether to store multi-dimensional data in row-major (C-style) or column-major (Fortran-style) order in memory.

Returns ------- out : ndarray Array of zeros with the given shape, dtype, and order.

See Also -------- zeros_like : Return an array of zeros with shape and type of input. empty : Return a new uninitialized array. ones : Return a new array setting values to one. full : Return a new array of given shape filled with value.

Examples -------- >>> np.zeros(5) array( 0., 0., 0., 0., 0.)

>>> np.zeros((5,), dtype=int) array(0, 0, 0, 0, 0)

>>> np.zeros((2, 1)) array([ 0.], [ 0.])

>>> s = (2,2) >>> np.zeros(s) array([ 0., 0.], [ 0., 0.])

>>> np.zeros((2,), dtype=('x', 'i4'), ('y', 'i4')) # custom dtype array((0, 0), (0, 0), dtype=('x', '<i4'), ('y', '<i4'))

Zero-pole-gain representation to state-space representation

Parameters ---------- z, p : sequence Zeros and poles. k : float System gain.

Returns ------- A, B, C, D : ndarray State space representation of the system, in controller canonical form.

Return polynomial transfer function representation from zeros and poles

Parameters ---------- z : array_like Zeros of the transfer function. p : array_like Poles of the transfer function. k : float System gain.

Returns ------- b : ndarray Numerator polynomial coefficients. a : ndarray Denominator polynomial coefficients.