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Function in discrete mathematics
In mathematics, the discrete Fourier transform (DFT) is a discrete version of the Fourier transform that converts a finite sequence of numbers into another
Discrete_Fourier_transform
Transform in numerical harmonic analysis
analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key
Discrete_wavelet_transform
Technique used in signal processing and data compression
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies
Discrete_cosine_transform
Discrete Fourier transform algorithm
Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT), or its inverse (IDFT), of a sequence. A Fourier transform converts
Fast_Fourier_transform
Mathematical transform using in signal processing
The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being
Modified discrete cosine transform
Modified_discrete_cosine_transform
Fourier analysis technique applied to sequences
mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values. The DTFT is
Discrete-time Fourier transform
Discrete-time_Fourier_transform
Mathematical transform on discrete signals
processing, discrete transforms (or discrete integral transform) are mathematical transforms, often linear transforms, of signals between discrete domains
Discrete_transform
Transform in mathematics
In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real
Discrete_sine_transform
Branch of mathematics
large numbers. The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms
Fourier_analysis
Fourier-related mathematical transform
A discrete Hartley transform (DHT) is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform (DFT), with analogous
Discrete_Hartley_transform
Integral transform and linear operator
similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform. The Hilbert transform was a motivating example for Antoni
Hilbert_transform
Linear transform from the time domain to the frequency domain
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex
Z-transform
Fourier-related transform for signals that change over time
Fourier transform times a window function, the STFT is also called windowed Fourier transform or time-dependent Fourier transform. In the discrete time case
Short-time_Fourier_transform
Signal representation
discrete rather than continuous. For example, the discrete Fourier transform maps a function having a discrete time domain into one having a discrete
Frequency_domain
transform Modified discrete cosine transform Discrete Hartley transform Discrete sine transform Discrete wavelet transform Hadamard transform (or, Walsh–Hadamard
List_of_transforms
Generalisation of Fourier transform to any ring
In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
Change of basis applied in quantum computing
quantum Fourier transform (QFT) is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier transform. The quantum Fourier
Quantum_Fourier_transform
applied mathematics, a discrete Chebyshev transform (abbreviated DCT, DChT, or DTT) is an analog of the discrete Fourier transform for a function of a real
Discrete_Chebyshev_transform
Signal processing operation
bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time
Bilinear_transform
Concept in applied mathematics
non-uniform discrete Fourier transform (NUDFT or NDFT) of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier
Non-uniform discrete Fourier transform
Non-uniform_discrete_Fourier_transform
Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients
List of Fourier-related transforms
List_of_Fourier-related_transforms
Mathematical function used in signal processing
{\operatorname {sinc} (Lf)}{(1-L^{2}f^{2})}}.\end{aligned}}} The discrete-time Fourier transform (DTFT) of the N + 1 {\displaystyle N+1} length, time-shifted
Hann_function
Involutive change of basis in linear algebra
themselves are purely real). The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent
Hadamard_transform
Mathematical technique used in data compression and analysis
Chirplet transform Complex wavelet transform Constant-Q transform Continuous wavelet transform Daubechies wavelet Discrete wavelet transform DjVu format
Wavelet_transform
In signal processing, a lapped transform is a type of linear discrete block transformation where the basis functions of the transformation overlap the
Lapped_transform
Fourier transform is a type of fast Fourier transform algorithm over finite fields. This algorithm first decomposes a discrete Fourier transform into several
Cyclotomic fast Fourier transform
Cyclotomic_fast_Fourier_transform
Study of discrete mathematical structures
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a one-to-one
Discrete_mathematics
Mathematical transform that expresses a function of time as a function of frequency
original Fourier transform on R or Rn, notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group =
Fourier_transform
Integral transform closely related to the Fourier transform
numbers) and of being its own inverse. The discrete version of the transform, the discrete Hartley transform (DHT), was introduced by Ronald N. Bracewell
Hartley_transform
Generalization of the discrete Fourier transform
Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. The Fourier transform of a
Fourier transform on finite groups
Fourier_transform_on_finite_groups
Conversion of continuous functions into discrete counterparts
applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This
Discretization
Discrete fourier transform expressed as a matrix
mathematics, a DFT matrix is a square matrix as an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal
DFT_matrix
Recursive algorithm in applied mathematics
In applied mathematics, the sliding discrete Fourier transform is a recursive algorithm to compute successive STFTs of input data frames that are a single
Sliding_DFT
Variant of fast Fourier transform
In mathematics, the irrational base discrete weighted transform (IBDWT) is a variant of the fast Fourier transform using an irrational base; it was developed
Irrational base discrete weighted transform
Irrational_base_discrete_weighted_transform
Mathematical operation
a discrete set (for example, the integers or a finite subset of integers). The Zak transform is a generalization of the discrete Fourier transform. The
Zak_transform
Series in digital signal processing
discrete sequence. So a DFS is a representation of one sequence in terms of another sequence. Well known examples are the Discrete Fourier transform and
Discrete_Fourier_series
Indian-American electrical engineer and computer scientist (born 1940)
University of New Mexico (UNM). He is best known for inventing the discrete cosine transform (DCT) in the early 1970s. The DCT is the most widely used data
Nasir_Ahmed_(engineer)
Indian-American electrical engineer (1931 - 2021)
co-invention of discrete cosine transform (DCT), along with Nasir Ahmed and T. Natarajan due to their landmark publication, Discrete Cosine Transform. Rao received
K._R._Rao
Integral transform useful in probability theory, physics, and engineering
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable
Laplace_transform
Special case of the short-time Fourier transform
The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency
Gabor_transform
Integral expressing the amount of overlap of one function as it is shifted over another
the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution
Convolution
Mathematical analysis of frequency content of signals
representation to a frequency domain representation. The discrete-domain multidimensional Fourier transform (FT) can be computed as follows: F ( w 1 , w 2 , …
Multidimensional_transform
In mathematics the finite Fourier transform may refer to either another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g
Finite_Fourier_transform
Mathematical signal manipulation by computers
frequency response. Bilinear transform Discrete Fourier transform Discrete-time Fourier transform Fast Fourier transform Filter design Goertzel algorithm
Digital_signal_processing
Variant Fourier transforms
terms which are summed. Discrete cosine transform Discrete sine transform Hartley transform — a composite sine & cosine transform that does not require
Sine_and_cosine_transforms
Mathematical algorithm
first approximation s ( J ) {\displaystyle s^{(J)}} . For the discrete wavelet transform (DWT), one computes recursively, starting with the coefficient
Fast_wavelet_transform
Mathematical algorithm
The chirp Z-transform (CZT) is a generalization of the discrete Fourier transform (DFT). While the DFT samples the Z plane at uniformly-spaced points
Chirp_Z-transform
Function for integral Fourier-like transform
related to harmonic analysis. Discrete wavelet transform (continuous in time) of a discrete-time (sampled) signal by using discrete-time filterbanks of dyadic
Wavelet
Mathematical operation
z-transform, and in particular for the case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying
Fractional_Fourier_transform
Duality for locally compact abelian groups
function on a finite abelian group can be recovered from its discrete Fourier transform. The theory, introduced by Lev Pontryagin and combined with the
Pontryagin_duality
The Mojette transform is an application of discrete geometry. More specifically, it is a discrete and exact version of the Radon transform, thus a projection
Mojette_transform
Short-time Fourier transform with variable resolution
resolution. The direct calculation of the constant-Q transform (either using naive discrete Fourier transform or slightly faster Goertzel algorithm) is slow
Constant-Q_transform
Function used in signal processing
that there is no leakage at a discrete set of harmonically-related frequencies sampled by the discrete Fourier transform (DFT). (The spectral nulls are
Window_function
wavelet transform (SWT) is a wavelet transform algorithm designed to overcome the lack of translation-invariance of the discrete wavelet transform (DWT)
Stationary_wavelet_transform
Data compression
signal processing. The most widely used transform coding technique in this regard is the discrete cosine transform (DCT), proposed by Nasir Ahmed in 1972
Transform_coding
Mathematical operation
example, in the context of the discrete-time Fourier transform (DTFT). In particular, the DTFT of the product of two discrete sequences is the periodic convolution
Circular_convolution
Theorem in mathematics
discrete-time Fourier transform (DTFT) operator. Consider two sequences u [ n ] {\displaystyle u[n]} and v [ n ] {\displaystyle v[n]} with transforms
Convolution_theorem
Type of filter in signal processing
{\displaystyle {\mathcal {F}}^{-1}} respectively denote the discrete-time Fourier transform (DTFT) and its inverse. Therefore, the complex-valued, multiplicative
Finite_impulse_response
System with a countable number of states
In theoretical computer science, a discrete system is a system with a countable number of states. Discrete systems may be contrasted with continuous systems
Discrete_system
Decomposition of periodic functions
compression standard uses the two-dimensional discrete cosine transform, a discrete form of the Fourier cosine transform, which uses only cosine as the basis function
Fourier_series
Technique to analyze the infrared spectrum of matter
material. Discrete Fourier transform – Function in discrete mathematics − for computing periodicity in evenly spaced data Fourier transform – Mathematical
Fourier-transform infrared spectroscopy
Fourier-transform_infrared_spectroscopy
Time-frequency transform in geophysics
\alpha t}\,d\alpha } . Discrete-time S-transform From the spectrum form of S-transform, we can derive the discrete-time S-transform. Let t = n Δ T f = m
S_transform
The modulated complex lapped transform (MCLT) is a lapped transform, similar to the modified discrete cosine transform, that explicitly represents the
Modulated complex lapped transform
Modulated_complex_lapped_transform
Mathematical model which is both linear and time-invariant
transfer function, which for a continuous-time or discrete-time system is the Laplace transform or Z-transform of the system's impulse response, respectively
Linear_time-invariant_system
Orthogonal wavelets
Ingrid Daubechies, are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for
Daubechies_wavelet
Discrete Fourier transform algorithm
The sparse Fourier transform (SFT) is a kind of discrete Fourier transform (DFT) for handling big data signals. Specifically, it is used in GPS synchronization
Sparse_Fourier_transform
Legendre polynomials are filtered out. Jerri, A.J. (1992). Integral and discrete transforms with applications and error analysis. Pure and Applied Mathematics
Finite_Legendre_transform
Effect in signal processing
function and then a discrete Fourier transform (DFT). But the DFT provides only a sparse sampling of the actual discrete-time Fourier transform (DTFT) spectrum
Spectral_leakage
Software library for computing discrete Fourier transforms
The Fastest Fourier Transform in the West (FFTW) is a software library for computing discrete Fourier transforms (DFTs) developed by Matteo Frigo and Steven
Fastest Fourier Transform in the West
Fastest_Fourier_Transform_in_the_West
Indicator function of positive numbers
the definition of H[0] is significant. The discrete-time unit impulse is the first difference of the discrete-time step: δ [ n ] = H [ n ] − H [ n − 1 ]
Heaviside_step_function
Multidimensional fast Fourier transform algorithm
dimensions by arbitrary radices. It breaks a multidimensional (MD) discrete Fourier transform (DFT) down into successively smaller MD DFTs until, ultimately
Vector-radix_FFT_algorithm
Compact encoding of digital data
Most forms of lossy compression are based on transform coding, especially the discrete cosine transform (DCT). It was first proposed in 1972 by Nasir
Data_compression
Sufficiency theorem for reconstructing signals from samples
Fourier transform that is zero outside of a finite region of frequencies. Intuitively we expect that when one reduces a continuous function to a discrete sequence
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Discrete Fourier transform for prime sizes
MIT Lincoln Laboratory, is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of prime sizes by re-expressing the
Rader's_FFT_algorithm
Basic method for pseudo-random number sampling
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, or the Smirnov
Inverse_transform_sampling
Mathematical function for the probability a given outcome occurs in an experiment
numbers), it is common to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability
Probability_distribution
Property of many linear time-invariant (LTI) systems
to the s-plane. When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly
Infinite_impulse_response
Analog of the continuous Laplace operator
mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For
Discrete_Laplace_operator
Video compression format, succeeds H.264/MPEG-4 AVC
uses the integer discrete cosine transform (DCT) with 4×4 and 8×8 block sizes, HEVC uses both integer DCT and discrete sine transform (DST) with varied
High_Efficiency_Video_Coding
transform (HFFT) is a tool in image and signal processing which uses fast Fourier transform (FFT) routines to compute the discrete Fourier transform (DFT)
Hexagonal fast Fourier transform
Hexagonal_fast_Fourier_transform
Theorem relating stationary processes' autocorrelations and power spectra
for the Fourier transform and Fourier inversion do not make sense. Further complicating the issue is that the discrete Fourier transform always exists for
Wiener–Khinchin_theorem
Mathematical function
order. This is the discrete analog of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous
Gaussian_function
short-time Fourier transform and the continuous wavelet transform. It can be expressed in terms of repeated Fourier transforms, and its discrete analogue can
Harmonic_wavelet_transform
Filter conversion technique
MZT, is a technique for converting a continuous-time filter design to a discrete-time filter (digital filter) design. The method works by mapping all poles
Matched_Z-transform_method
Very general problem in computer science
computer science. It is a generalization of problems including factoring, discrete logarithm, graph isomorphism, and the shortest vector problem. This makes
Hidden_subgroup_problem
Digitally coded format for audio signals
AAC. Discrete cosine transform (DCT), developed by Nasir Ahmed, T. Natarajan and K. R. Rao in 1974, provided the basis for the modified discrete cosine
Audio_coding_format
applied mathematics, the starred transform, or star transform, is a discrete-time variation of the Laplace transform, so-named because of the asterisk
Starred_transform
Distortion of media caused by lossy data compression
common digital compression artifacts are DCT blocks, caused by the discrete cosine transform (DCT) compression algorithm used in many digital media standards
Compression_artifact
Type of data compression
needs). The most widely used lossy compression algorithm is the discrete cosine transform (DCT), first published by Nasir Ahmed, T. Natarajan and K. R.
Lossy_compression
Varying physical quantity that conveys information
might transform from the time domain to a frequency or s domain; or from discrete time (n) to frequency or z domains. Systems can also be transformed between
Signal
_{k=1}^{N}{A_{k}e^{s_{k}nT}u[n]}\,} Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function H ( z ) = T ∑
Impulse_invariance
Concept relating to waves and signals
spectrum domain (such as frequency or wavelength) or discrete if it attains non-zero values only in a discrete set over the independent variable, with band gaps
Spectrum_(physical_sciences)
transform, the large number of computations necessary for the discrete Fourier Transform motivated researchers to reduce the number of calculations required
Blackman–Tukey_transformation
Filter in electronics and signal processing
arbitrary discrete filter kernel using the Fast Fourier transform (FFT), and can be applied for Gaussian filtering. Since the Fourier transform of the Gaussian
Gaussian_filter
Linear algebra matrix
diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform. They can be
Circulant_matrix
Mathematical operation
567.1 of. The expansion coefficients fm,t are accessible with discrete Fourier transform techniques: if the radial distance is scaled with r / R ≡ sin
Hankel_transform
Reduction of image size to save storage and transmission costs
Methods for lossy compression: Transform coding – This is the most commonly used method. Discrete Cosine Transform (DCT) – The most widely used form
Image_compression
Theorem in mathematics
self-dual and the Pontryagin–Fourier transform is what is called discrete Fourier transform in applied contexts. Parseval's theorem can also be expressed
Parseval's_theorem
Diagram showing the singularities of a given control system's transfer function
represent either a continuous-time or a discrete-time system: Continuous-time systems use the Laplace transform and are plotted in the s-plane: s = σ +
Pole–zero_plot
Computation process in mathematical algorithms
fast Fourier transform algorithms, a butterfly is a portion of the computation that combines the results of smaller discrete Fourier transforms (DFTs) into
Butterfly_diagram
DISCRETE TRANSFORM
DISCRETE TRANSFORM
Girl/Female
Greek
The laurel tree. The mythological virtuous Daphne was transformed into a laurel tree to protect...
Girl/Female
Greek
Most beautiful. In Mythology the Arcadian nymph Calista transformed into a she-bear; then into...
Girl/Female
Arabic, Muslim
Firm; Energetic; Judicious; Discreet; Prudent; Feminine of Hazim
Girl/Female
Australian, Vietnamese
Discreet Grace
Boy/Male
Muslim/Islamic
Discreet prudent
Boy/Male
Afghan, Arabic, Australian, German, Muslim
Thunder; Keen; Sharp; Discreet; Prudent
Girl/Female
Greek
Bee. Famous bearer: Melissa, Mythological princess of Crete transformed to a bee after learning...
Surname or Lastname
English and French
English and French : regional name from Old French Poitevin, denoting someone from Poitou in western France. The form Potvin has long been established in England and was brought to the U.S. from there. However, French bearers of the surname Poitevin also came to the New World, where their surname underwent a similar transformation on arrival in New England.
Girl/Female
Greek
Most beautiful. Calista was a Mythological Arcadian who transformed into a she-bear, then into...
Girl/Female
Greek American
Bee. Famous bearer: Melissa, Mythological princess of Crete transformed to a bee after learning...
Girl/Female
Greek American
Bee. Famous bearer: Melissa, Mythological princess of Crete transformed to a bee after learning...
Girl/Female
Greek
Bee. Famous bearer: Melissa, Mythological princess of Crete transformed to a bee after learning...
Surname or Lastname
English
English : habitational name from Lichfield in Staffordshire. The first element preserves a British name recorded as Letocetum during the Romano-British period. This means ‘gray wood’, from words which are the ancestors of Welsh llŵyd ‘gray’ and coed ‘wood’. By the Old English period this had been reduced to Licced, and the element feld ‘pasture’, ‘open country’ was added to describe a patch of cleared land within the ancient wood.English : habitational name from Litchfield in Hampshire, recorded in Domesday Book as Liveselle. This is probably from an Old English hlīf ‘shelter’ + Old English scylf ‘shelf’, ‘ledge’. The subsequent transformation of the place name may be the result of folk etymological association with Old English hlið, hlid ‘slope’ + feld ‘open country’.
Girl/Female
Greek
Bee. Famous bearer: Melissa, Mythological princess of Crete transformed to a bee after learning...
Girl/Female
Greek
Bee. Famous bearer: Melissa, Mythological princess of Crete transformed to a bee after learning...
Girl/Female
Latin
or Selena. One of seven mythological daughters of Atlas transformed by Zeus into stars of the...
Girl/Female
Latin
or Selena. One of seven mythological daughters of Atlas transformed by Zeus into stars of the...
Girl/Female
Latin
or Selena. One of seven mythological daughters of Atlas transformed by Zeus into stars of the...
Girl/Female
Bengali, Hindu, Indian
Discreet; Enrich; Impressive; Advantage
Girl/Female
Israeli
The laurel tree. The mythological virtuous Daphne was transformed into a laurel tree to protect...
DISCRETE TRANSFORM
DISCRETE TRANSFORM
Female
Hindi/Indian
(Bengali নীলা): Hindi name NILA means "sky blue."
Surname or Lastname
English (Kent)
English (Kent) : unexplained.perhaps an Americanized form of German Engemeyer, a topographic name for a tenant farmer who lived in a narrow place, i.e. a deep, narrow valley, from eng ‘narrow’ (see Enge) + Meyer ‘tenant farmer’.
Boy/Male
Arabic, Indian, Muslim
A Prophet's Name
Girl/Female
Arabic, Muslim
Happiness
Boy/Male
Christian & English(British/American/Australian)
Chief
Surname or Lastname
English
English : occupational name for a trumpeter, Middle English bemere (Old English bēmere, bīemere).Americanized spelling of German Boehmer or Bäumer (see Baumer).
Girl/Female
Tamil
Garland of victory
Boy/Male
Indian, Punjabi, Sikh, Traditional
Brave and Divine in Knowledge
Boy/Male
Indian, Punjabi, Sikh
Remembering the Lord by Heart
Boy/Male
American, Australian, British, Celtic, Chinese, Christian, Danish, English, French, German, Gujarati, Hindu, Indian, Irish
Champion; Blue; Lord Shiva (Blue Throat); Engineer to the Gods with Twin Nal Helped Rama Build the Bridge to Lanka
DISCRETE TRANSFORM
DISCRETE TRANSFORM
DISCRETE TRANSFORM
DISCRETE TRANSFORM
DISCRETE TRANSFORM
n.
A decree.
a.
Not prudent; not discreet; ill advised.
a.
Marking distinction or separation; disjunctive.
superl.
Possessed of discernment, especially in avoiding error or evil, and in the adaptation of means to ends; prudent; sagacious; judicious; not rash or heedless; cautious.
a.
Separate; not coalescent; -- said of things usually coalescent.
a.
Not compact; not having the parts firmly united; not solid; incoherent; loose; discrete.
a.
Miscreated; illegitimate; forged; as, miscreate titles.
v. t.
To separate.
a.
Separate; distinct; disjunct.
a.
Not discrete or separated; compact; homogenous.
a.
Indiscreet.
superl.
Differing; distinct.
v. t.
To describe.
adv.
Separately; disjunctively.
a.
Directed or governed by sound judgment; having sound judgment; wise; prudent; sagacious; discreet.
a.
Not discreet; wanting in discretion.
a.
Disjunctive; containing a disjunctive or discretive clause; as, "I resign my life, but not my honor," is a discrete proposition.
v. t.
To reduce to a lower rating or rank; to degrade.
adv.
In a discretive manner.
v. t.
To create badly or amiss.