Method and an apparatus for fast convolution of signals with a one-sided exponential function

A method for fast convolution of a signal with complex exponential function localized around an origin by an exponential function envelope minimizing memory requirement, comprising:

• modulating the input signal with the complex exponential function;
• accumulating into a register a number of the modulated signal samples equal to a pre-determined number W of samples of a truncated one-sided exponential function;
• splitting the modulated signal into a real part and an imaginary part;
• carrying out a fast convolution of the real part of the modulated signal with the exponential function envelope to obtain a first convolved signal;
• carrying out a fast convolution of the imaginary part of the modulated signal with the exponential function envelope to obtain a second convolved signal;
• merging the first convolved signal and the second convolved signal;
• and
• demodulating the merged signal with the complex exponential function.

An apparatus for fast convolution of a signal with a one-sided exponential function minimizing memory requirement, comprising:

• an hardware accumulator comprising a register configured to accumulate N samples of the signal, wherein N equals a pre-determined number W of samples of a truncated one-sided exponential function;
• a first multiplier configured to multiply a value in the hardware accumulator by an inverse value of a decay of the exponential function;
• a second multiplier configure to multiply a value of a first sample of the signal W samples ahead from a second sample of the signal by the value of the decay of the exponential function raised to the power of the number W;
• and
• a summer configured to
• sum the multiplied value in the hardware accumulator with the multiplied value of the first sample of the signal and with a negative value of the second sample of the signal,
• provide the summed value to the hardware accumulator, and
• output the summed value.

An apparatus for fast convolution of a signal with complex exponential function localized around an origin by an exponential function envelope minimizing memory requirement, comprising:

• a hardware modulator configured to modulate the signal with the complex exponential function;
• a hardware accumulator comprising a register configured to accumulate N samples of the signal, wherein N equals a pre-determined number W of samples of a truncated one-sided exponential function;
• a hardware splitter configured to split the modulated signal into a real part and an imaginary part;
• and
• a hardware processing unit configured to
• carry out a fast convolution of the real part of the modulated signal with the exponential function envelope to obtain a first convolved signal, and
• carry out a fast convolution of the imaginary part of the modulated signal with the exponential function envelope to obtain a second convolved signal;
• a hardware means for merging the first convolved signal and the second convolved signal;
• and
• a hardware demodulator configured to demodulate the merged signal with the complex exponential function.