Measuring Waveforms With The Digital Infinite Exponential Transform

A method, performed by one or more processing devices, of processing a signal of interest from a compound signal, the method comprising:

• generating a matrix wherein each column is associated with a corresponding frequency bin of a plurality of frequency bins, each frequency bin extending over a portion of a bandwidth of the compound signal;
• initializing a first row of the matrix to zero;
• obtaining the compound signal in the form of a digital signal, with sampling rate S samples per second, from data storage or from conversion of a physical analog signal received from an analog source or captured by a physical analog device;
• for each sample of the digital signal, recursively computing, using a processing device, a new entry for the matrix, wherein the computing comprises, for each frequency bin:
• multiplying a previous row entry of a given frequency bin by r, where r is a constant greater than zero and less than 1, and
• adding a current digital signal sample multiplied by a real constant and by e−i2πnf/S, where f is a center frequency of the given frequency bin, S is the sampling rate, and n is a current sample number;
• and
• at least one of storing the matrix, transmitting the matrix, displaying the matrix, and identifying the signal of interest from the matrix, whereby an uncertainty of which frequency bin the signal of interest exists in is eliminated.

A method, performed by one or more processing devices, of processing a signal of interest from a compound signal, the method comprising:

• generating an array wherein each element of the array is associated with a corresponding frequency bin of a plurality of frequency bins, each frequency bin extending over a portion of a bandwidth of the compound signal;
• initializing the array to zero;
• obtaining the compound signal in the form of a digital signal, with sampling rate S samples per second, from data storage or from conversion of a physical analog signal received from an analog source or captured by a physical analog device;
• for each sample of the digital signal, recursively updating the array by:
• for each frequency bin, where f is the center frequency of the bin:
• multiplying the previous value for that bin in the array by the complex number r*ei2πf/S, where r is an exponential decay constant greater than zero and less than 1, f is the center frequency of the bin, and S is the sampling rate;
• and
• adding the new signal sample multiplied by a real constant;
• saving this updated array as a new row in a matrix;
• and
• at least one of storing the matrix, transmitting the matrix, displaying the matrix, and identifying the signal of interest from the matrix, whereby an uncertainty of which frequency bin the signal of interest exists in is eliminated.

A method, performed by one or more processing devices, of processing a signal of interest from a compound signal, the method comprising:

• generating an array wherein each element of the array is associated with a corresponding frequency bin of a plurality of frequency bins, each frequency bin extending over a portion of a bandwidth of the compound signal;
• initializing the array to zero;
• obtaining the compound signal in the form of a digital signal, with sampling rate S samples per second, from data storage or from conversion of a physical analog signal received from an analog source or captured by a physical analog device;
• for each sample of the digital signal, recursively updating the array by:
• for each frequency bin, where f is the center frequency of the bin:
• multiplying the previous value for that bin in the array by r, where r is an exponential decay constant greater than zero and less than 1;
• and
• adding the new signal sample multiplied by a real constant and by e−i2πnf/S, where f is a center frequency of the given frequency bin, S is the sampling rate, and n is a current sample number;
• saving this updated array as a new row in a matrix;
• and
• at least one of storing the matrix, transmitting the matrix, displaying the matrix, and identifying the signal of interest from the matrix, whereby an uncertainty of which frequency bin the signal of interest exists in is eliminated.

A method, performed by one or more processing devices, of processing a signal of interest from a compound signal, the method comprising:

• generating a matrix wherein each column is associated with a corresponding frequency bin of a plurality of frequency bins, each frequency bin extending over a portion of a bandwidth of the compound signal;
• initializing a first row of the matrix to zero;
• obtaining the compound signal in the form of a digital signal, with sampling rate S samples per second, from data storage or from conversion of a physical analog signal received from an analog source or captured by a physical analog device;
• performing a Fourier transform on the digital signal to populate each cell of the matrix with values;
• multiplying the value in each cell of the matrix by the complex number e−i2πnf/S, where n corresponds to the n'th sample in the matrix, f is the center frequency of the bin, and S is the sampling rate;
• and
• at least one of storing the matrix, transmitting the matrix, displaying the matrix, and identifying the signal of interest from the matrix, whereby an uncertainty of which frequency bin the signal of interest exists in is eliminated.