Time and Frequency Domain Representation

Notation and basic equations Following the representation in [75], the notation and the basic equations are as follows. 

Continuous functions and signals in the time domain are denoted by lower case letters with the argument in parentheses, e.g. x(t). Sampling at constant intervals A t produces a discrete approximation x[n] to the continuous signal, defined at times f = n A t, n = 0,1,2. Square brackets are used for the arguments of discrete functions. The Fourier transform establishes the connection between the time and frequency domains [76]  

Frequency domain functions are denoted by upper case letters. Of importance for the TDFRS experiment is the discrete Fourier transform of an array of N data points within a period of N At  

The response y(t) of a linear system to an excitation x(t) is a convolution of x(t) with the response function h(t). The TDFRS experiments are performed with periodic boundary conditions and discrete sampling, asking for the discrete periodic convolution 

The convolution theorem reduces Eq. (57) to a simple product in frequency space Y = XH (58) 

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To understand the basic concepts of modulation, we first review time- and frequency-domain representations of signals. The information present in m(t) can be completely specified by a complex function of speech amplitude vs. frequency, M f), obtained using the Fourier transform. Since M f) contains aU of the information in m t), it is possible to go back and forth between m t) and M f) using the forward and inverse Fourier transforms. 

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