X-Git-Url: http://git.kpe.io/?a=blobdiff_plain;f=libctsim%2Ffilter.cpp;h=640bfb2d6b84783904d5e42034052ee4cabc7845;hb=b847a34057751df3a3f4e2bf795227a13ef4788c;hp=c62eb24330d59e0c55645e6918977bc73f5d726a;hpb=2d39e823ba389fc68e5317c422b55be006094252;p=ctsim.git diff --git a/libctsim/filter.cpp b/libctsim/filter.cpp index c62eb24..640bfb2 100644 --- a/libctsim/filter.cpp +++ b/libctsim/filter.cpp @@ -1,5 +1,5 @@ /***************************************************************************** -** FILE IDENTIFICATION +** File IDENTIFICATION ** ** Name: filter.cpp ** Purpose: Routines for signal-procesing filters @@ -9,7 +9,7 @@ ** This is part of the CTSim program ** Copyright (C) 1983-2000 Kevin Rosenberg ** -** $Id: filter.cpp,v 1.3 2000/06/22 10:17:28 kevin Exp $ +** $Id: filter.cpp,v 1.10 2000/07/05 01:34:46 kevin Exp $ ** ** This program is free software; you can redistribute it and/or modify ** it under the terms of the GNU General Public License (version 2) as @@ -32,88 +32,205 @@ * SignalFilter::SignalFilter Construct a signal * * SYNOPSIS - * f = SignalFilter (filt_type, bw, xmin, xmax, n, param, domain, analytic) + * f = SignalFilter (filt_type, bw, filterMin, filterMax, n, param, domain, analytic) * double f Generated filter vector * int filt_type Type of filter wanted * double bw Bandwidth of filter - * double xmin, xmax Filter limits - * int n Number of points in filter + * double filterMin, filterMax Filter limits + * int nSignalPoints Number of points in signal * double param General input parameter to filters - * int domain FREQ or SPATIAL domain wanted + * int domain FREQUENCY or SPATIAL domain wanted * int numint Number if intervals for calculating discrete inverse fourier xform * for spatial domain filters. For ANALYTIC solutions, use numint = 0 */ -SignalFilter::SignalFilter (const char* filterName, double bw, double xmin, double xmax, int n, double param, const char* domainName, int numint) +SignalFilter::SignalFilter (const char* filterName, const char* filterMethodName, double bw, double signalIncrement, int nSignalPoints, double param, const char* domainName, int numIntegral = 0) { + m_vecFilter = NULL; + m_vecFourierCosTable = NULL; + m_vecFourierSinTable = NULL; m_idFilter = convertFilterNameToID (filterName); + if (m_idFilter == FILTER_INVALID) { + m_fail = true; + m_failMessage = "Invalid Filter name "; + m_failMessage += filterName; + return; + } + m_idFilterMethod = convertFilterMethodNameToID (filterMethodName); + if (m_idFilterMethod == FILTER_METHOD_INVALID) { + m_fail = true; + m_failMessage = "Invalid filter method name "; + m_failMessage += filterMethodName; + return; + } m_idDomain = convertDomainNameToID (domainName); - init (m_idFilter, bw, xmin, xmax, n, param, m_idDomain, numint); + if (m_idDomain == DOMAIN_INVALID) { + m_fail = true; + m_failMessage = "Invalid domain name "; + m_failMessage += domainName; + return; + } + init (m_idFilter, m_idFilterMethod, bw, signalIncrement, nSignalPoints, param, m_idDomain, numIntegral); } -SignalFilter::SignalFilter (const FilterID filterID, double bw, double xmin, double xmax, int n, double param, const DomainID domainID, int numint) +SignalFilter::SignalFilter (const FilterID filterID, const FilterMethodID filterMethodID, double bw, double signalIncrement, int nSignalPoints, double param, const DomainID domainID, int numIntegral = 0) { - init (filterID, bw, xmin, xmax, n, param, domainID, numint); + init (filterID, filterMethodID, bw, signalIncrement, nSignalPoints, param, domainID, numIntegral); +} + +SignalFilter::SignalFilter (const char* filterName, const char* domainName, double bw, double param, int numIntegral = 0) +{ + m_bw = bw; + m_nSignalPoints = 0; + m_nFilterPoints = 0; + m_vecFilter = NULL; + m_vecFourierCosTable = NULL; + m_vecFourierSinTable = NULL; + m_filterParam = param; + m_numIntegral = numIntegral; + m_idFilter = convertFilterNameToID (filterName); + if (m_idFilter == FILTER_INVALID) { + m_fail = true; + m_failMessage = "Invalid Filter name "; + m_failMessage += filterName; + return; + } + m_idDomain = convertDomainNameToID (domainName); + if (m_idDomain == DOMAIN_INVALID) { + m_fail = true; + m_failMessage = "Invalid domain name "; + m_failMessage += domainName; + return; + } } void -SignalFilter::init (const FilterID filterID, double bw, double xmin, double xmax, int n, double param, const DomainID domainID, int numint) +SignalFilter::init (const FilterID filterID, const FilterMethodID filterMethodID, double bw, double signalIncrement, int nSignalPoints, double param, const DomainID domainID, int numint) { m_bw = bw; m_idFilter = filterID; m_idDomain = domainID; - if (m_idFilter == FILTER_INVALID || m_idDomain == DOMAIN_INVALID) { + m_idFilterMethod = filterMethodID; + if (m_idFilter == FILTER_INVALID || m_idDomain == DOMAIN_INVALID || m_idFilterMethod == FILTER_METHOD_INVALID) { m_fail = true; return; } + m_traceLevel = TRACE_NONE; m_nameFilter = convertFilterIDToName (m_idFilter); m_nameDomain = convertDomainIDToName (m_idDomain); + m_nameFilterMethod = convertFilterMethodIDToName (m_idFilterMethod); m_fail = false; - m_nPoints = n; - m_xmin = xmin; - m_xmax = xmax; - m_vecFilter = new double[n]; - - double xinc = (m_xmax - m_xmin) / (m_nPoints - 1); - - if (m_idFilter == FILTER_SHEPP) { - double a = 2 * m_bw; - double c = - 4. / (a * a); - int center = (m_nPoints - 1) / 2; - int sidelen = center; - m_vecFilter[center] = 4. / (a * a); - - for (int i = 1; i <= sidelen; i++ ) - m_vecFilter [center + i] = m_vecFilter [center - i] = c / (4 * (i * i) - 1); - } else if (m_idDomain == DOMAIN_FREQ) { - double x; - int i; - for (x = m_xmin, i = 0; i < m_nPoints; x += xinc, i++) - m_vecFilter[i] = frequencyResponse (x, param); - } else if (m_idDomain == DOMAIN_SPATIAL) { - double x; - int i; - for (x = m_xmin, i = 0; i < m_nPoints; x += xinc, i++) - if (numint == 0) - m_vecFilter[i] = spatialResponseAnalytic (x, param); - else - m_vecFilter[i] = spatialResponseCalc (x, param, numint); - } else { - sys_error (ERR_WARNING, "Illegal domain %d [filt_generate]", m_idDomain); + m_nSignalPoints = nSignalPoints; + m_signalInc = signalIncrement; + m_filterParam = param; + + if (m_idFilterMethod == FILTER_METHOD_FOURIER) { + int nFourier = m_nSignalPoints * m_nSignalPoints + 1; + double angleIncrement = (2. * PI) / m_nSignalPoints; + m_vecFourierCosTable = new double[ nFourier ]; + m_vecFourierSinTable = new double[ nFourier ]; + for (int i = 0; i < nFourier; i++) { + m_vecFourierCosTable[i] = cos (angleIncrement * i); + m_vecFourierSinTable[i] = sin (angleIncrement * i); + } + m_nFilterPoints = m_nSignalPoints; + m_filterMin = -1. / (2 * m_signalInc); + m_filterMax = 1. / (2 * m_signalInc); + m_filterInc = (m_filterMax - m_filterMin) / m_nFilterPoints; + m_vecFilter = new double [m_nFilterPoints]; + int halfFilter = m_nFilterPoints / 2; + for (int i = 0; i < halfFilter; i++) + m_vecFilter[i] = static_cast(i) / (halfFilter - 1) / (2 * m_signalInc); + for (int i = 0; i < halfFilter; i++) + m_vecFilter[m_nFilterPoints - i - 1] = static_cast(i) / (halfFilter - 1) / (2 * m_signalInc); + if (halfFilter % 2) // odd + m_vecFilter[halfFilter] = 1 / (2 * m_signalInc); + } else if (m_idFilterMethod == FILTER_METHOD_FFT || m_idFilterMethod == FILTER_METHOD_FFT_ZEROPAD_2 || m_idFilterMethod == FILTER_METHOD_FFT_ZEROPAD_4) { + m_nFilterPoints = m_nSignalPoints; + if (m_idFilterMethod == FILTER_METHOD_FFT_ZEROPAD_2 || m_idFilterMethod == FILTER_METHOD_FFT_ZEROPAD_4) { + double logBase2 = log(m_nSignalPoints) / log(2); + int nextPowerOf2 = static_cast(floor(logBase2)) + 1; + if (m_idFilterMethod == FILTER_METHOD_FFT_ZEROPAD_4) + nextPowerOf2++; + if (logBase2 != floor(logBase2)) + nextPowerOf2++; + m_nFilterPoints = 1 << nextPowerOf2; + cout << "nFilterPoints = " << m_nFilterPoints << endl; + } + m_filterMin = -1. / (2 * m_signalInc); + m_filterMax = 1. / (2 * m_signalInc); + m_filterInc = (m_filterMax - m_filterMin) / m_nFilterPoints; + m_vecFilter = new double [m_nFilterPoints]; + int halfFilter = m_nFilterPoints / 2; + for (int i = 0; i < halfFilter; i++) + m_vecFilter[i] = static_cast(i) / (halfFilter - 1) / (2 * m_signalInc); + for (int i = 0; i < halfFilter; i++) + m_vecFilter[m_nFilterPoints - i - 1] = static_cast(i) / (halfFilter - 1) / (2 * m_signalInc); + if (halfFilter % 2) // odd + m_vecFilter[halfFilter] = 1 / (2 * m_signalInc); + +#if HAVE_FFTW + m_planForward = fftw_create_plan (m_nFilterPoints, FFTW_FORWARD, FFTW_ESTIMATE); + m_planBackward = fftw_create_plan (m_nFilterPoints, FFTW_BACKWARD, FFTW_ESTIMATE); +#endif + } + + if (m_idFilterMethod == FILTER_METHOD_CONVOLUTION) { + m_nFilterPoints = 2 * m_nSignalPoints - 1; + m_filterMin = -m_signalInc * (m_nSignalPoints - 1); + m_filterMax = m_signalInc * (m_nSignalPoints - 1); + m_filterInc = (m_filterMax - m_filterMin) / (m_nFilterPoints - 1); + m_numIntegral = numint; + m_vecFilter = new double[ m_nFilterPoints ]; + + if (m_idFilter == FILTER_SHEPP) { + double a = 2 * m_bw; + double c = - 4. / (a * a); + int center = (m_nFilterPoints - 1) / 2; + int sidelen = center; + m_vecFilter[center] = 4. / (a * a); + + for (int i = 1; i <= sidelen; i++ ) + m_vecFilter [center + i] = m_vecFilter [center - i] = c / (4 * (i * i) - 1); + } else if (m_idDomain == DOMAIN_FREQUENCY) { + double x; + int i; + for (x = m_filterMin, i = 0; i < m_nFilterPoints; x += m_filterInc, i++) + m_vecFilter[i] = frequencyResponse (x, param); + } else if (m_idDomain == DOMAIN_SPATIAL) { + double x; + int i; + for (x = m_filterMin, i = 0; i < m_nFilterPoints; x += m_filterInc, i++) + if (numint == 0) + m_vecFilter[i] = spatialResponseAnalytic (x, param); + else + m_vecFilter[i] = spatialResponseCalc (x, param, numint); + } else { + m_failMessage = "Illegal domain name "; + m_failMessage += m_idDomain; m_fail = true; + } } } SignalFilter::~SignalFilter (void) { delete m_vecFilter; + delete m_vecFourierSinTable; + delete m_vecFourierCosTable; +#if HAVE_FFTW + if (m_idFilterMethod == FILTER_METHOD_FFT) { + fftw_destroy_plan(m_planForward); + fftw_destroy_plan(m_planBackward); + } +#endif } -SignalFilter::FilterID +const SignalFilter::FilterID SignalFilter::convertFilterNameToID (const char *filterName) { - FilterID filterID; + FilterID filterID = FILTER_INVALID; if (strcasecmp (filterName, FILTER_BANDLIMIT_STR) == 0) filterID = FILTER_BANDLIMIT; @@ -135,10 +252,6 @@ SignalFilter::convertFilterNameToID (const char *filterName) filterID = FILTER_ABS_COSINE; else if (strcasecmp (filterName, FILTER_SHEPP_STR) == 0) filterID = FILTER_SHEPP; - else { - sys_error(ERR_WARNING, "Invalid filter type %s\n", filterName); - filterID = FILTER_INVALID; - } return (filterID); } @@ -172,17 +285,57 @@ SignalFilter::convertFilterIDToName (const FilterID filterID) return (name); } +const SignalFilter::FilterMethodID +SignalFilter::convertFilterMethodNameToID (const char* const filterMethodName) +{ + FilterMethodID fmID = FILTER_METHOD_INVALID; + + if (strcasecmp (filterMethodName, FILTER_METHOD_CONVOLUTION_STR) == 0) + fmID = FILTER_METHOD_CONVOLUTION; + else if (strcasecmp (filterMethodName, FILTER_METHOD_FOURIER_STR) == 0) + fmID = FILTER_METHOD_FOURIER; + else if (strcasecmp (filterMethodName, FILTER_METHOD_FFT_STR) == 0) + fmID = FILTER_METHOD_FFT; + else if (strcasecmp (filterMethodName, FILTER_METHOD_FFT_ZEROPAD_2_STR) == 0) + fmID = FILTER_METHOD_FFT_ZEROPAD_2; + else if (strcasecmp (filterMethodName, FILTER_METHOD_FFT_ZEROPAD_4_STR) == 0) + fmID = FILTER_METHOD_FFT_ZEROPAD_4; + else if (strcasecmp (filterMethodName, FILTER_METHOD_FFT_ZEROPAD_6_STR) == 0) + fmID = FILTER_METHOD_FFT_ZEROPAD_6; + + return (fmID); +} + +const char * +SignalFilter::convertFilterMethodIDToName (const FilterMethodID fmID) +{ + const char *name = ""; + + if (fmID == FILTER_METHOD_CONVOLUTION) + return (FILTER_METHOD_CONVOLUTION_STR); + else if (fmID == FILTER_METHOD_FOURIER) + return (FILTER_METHOD_FOURIER_STR); + else if (fmID == FILTER_METHOD_FFT) + return (FILTER_METHOD_FFT_STR); + else if (fmID == FILTER_METHOD_FFT_ZEROPAD_2) + return (FILTER_METHOD_FFT_ZEROPAD_2_STR); + else if (fmID == FILTER_METHOD_FFT_ZEROPAD_4) + return (FILTER_METHOD_FFT_ZEROPAD_4_STR); + else if (fmID == FILTER_METHOD_FFT_ZEROPAD_6) + return (FILTER_METHOD_FFT_ZEROPAD_6_STR); + + return (name); +} + const SignalFilter::DomainID SignalFilter::convertDomainNameToID (const char* const domainName) { - DomainID dID; + DomainID dID = DOMAIN_INVALID; if (strcasecmp (domainName, DOMAIN_SPATIAL_STR) == 0) dID = DOMAIN_SPATIAL; - else if (strcasecmp (domainName, DOMAIN_FREQ_STR) == 0) - dID = DOMAIN_FREQ; - else - dID = DOMAIN_INVALID; + else if (strcasecmp (domainName, DOMAIN_FREQUENCY_STR) == 0) + dID = DOMAIN_FREQUENCY; return (dID); } @@ -194,28 +347,106 @@ SignalFilter::convertDomainIDToName (const DomainID domain) if (domain == DOMAIN_SPATIAL) return (DOMAIN_SPATIAL_STR); - else if (domain == DOMAIN_FREQ) - return (DOMAIN_FREQ_STR); + else if (domain == DOMAIN_FREQUENCY) + return (DOMAIN_FREQUENCY_STR); return (name); } +void +SignalFilter::filterSignal (const float input[], double output[]) const +{ + if (m_idFilterMethod == FILTER_METHOD_CONVOLUTION) { + for (int i = 0; i < m_nSignalPoints; i++) + output[i] = convolve (input, m_signalInc, i, m_nSignalPoints); + } else if (m_idFilterMethod == FILTER_METHOD_FOURIER) { + complex fftSignal[m_nSignalPoints]; + complex complexOutput[m_nSignalPoints]; + complex filteredSignal[m_nSignalPoints]; + finiteFourierTransform (input, fftSignal, m_nSignalPoints, -1); + if (m_traceLevel >= TRACE_PLOT) { + double test[m_nSignalPoints]; + for (int i = 0; i < m_nSignalPoints; i++) + test[i] = abs(fftSignal[i]); + ezplot_1d(test, m_nSignalPoints); + cio_kb_getc(); + } + dotProduct (m_vecFilter, fftSignal, filteredSignal, m_nSignalPoints); + if (m_traceLevel >= TRACE_PLOT) { + double test[m_nSignalPoints]; + for (int i = 0; i < m_nSignalPoints; i++) + test[i] = abs(filteredSignal[i]); + ezplot_1d(test, m_nSignalPoints); + cio_kb_getc(); + } + finiteFourierTransform (filteredSignal, complexOutput, m_nSignalPoints, 1); + for (int i = 0; i < m_nSignalPoints; i++) + output[i] = abs( complexOutput[i] ); + } else if (m_idFilterMethod == FILTER_METHOD_FFT || FILTER_METHOD_FFT_ZEROPAD_2 || FILTER_METHOD_FFT_ZEROPAD_4) { + fftw_complex in[m_nFilterPoints], out[m_nFilterPoints]; + for (int i = 0; i < m_nSignalPoints; i++) { + in[i].re = input[i]; + in[i].im = 0; + } + for (int i = m_nSignalPoints; i < m_nFilterPoints; i++) { + in[i].re = in[i].im = 0; // ZeroPad + } + fftw_one(m_planForward, in, out); + if (m_traceLevel >= TRACE_PLOT) { + double test[m_nFilterPoints]; + for (int i = 0; i < m_nFilterPoints; i++) + test[i] = sqrt(out[i].re * out[i].re + out[i].im * out[i].im); + ezplot_1d(test, m_nFilterPoints); + cio_kb_getc(); + } + for (int i = 0; i < m_nFilterPoints; i++) { + out[i].re = m_vecFilter[i] * out[i].re / m_nSignalPoints; + out[i].im = m_vecFilter[i] * out[i].im / m_nSignalPoints; + } + if (m_traceLevel >= TRACE_PLOT) { + double test[m_nFilterPoints]; + for (int i = 0; i < m_nFilterPoints; i++) + test[i] = sqrt(out[i].re * out[i].re + out[i].im * out[i].im); + ezplot_1d(test, m_nFilterPoints); + cio_kb_getc(); + } + fftw_one(m_planBackward, out, in); + if (m_traceLevel >= TRACE_PLOT) { + double test[m_nFilterPoints]; + for (int i = 0; i < m_nFilterPoints; i++) + test[i] = sqrt(in[i].re * in[i].re + in[i].im * in[i].im); + ezplot_1d(test, m_nFilterPoints); + cio_kb_getc(); + } + for (int i = 0; i < m_nSignalPoints; i++) + output[i] = sqrt (in[i].re * in[i].re + in[i].im * in[i].im); + } +} + double -SignalFilter::response (const char* filterName, const char* domainName, double bw, double x, double filt_param) +SignalFilter::response (double x) { double response = 0; - FilterID filterID = convertFilterNameToID (filterName); - DomainID domainID = convertDomainNameToID (domainName); - if (domainID == DOMAIN_SPATIAL) - response = spatialResponseAnalytic (filterID, bw, x, filt_param); - else if (domainID == DOMAIN_FREQ) - response = frequencyResponse (filterID, bw, x, filt_param); + if (m_idDomain == DOMAIN_SPATIAL) + response = spatialResponse (m_idFilter, m_bw, x, m_filterParam, m_numIntegral); + else if (m_idDomain == DOMAIN_FREQUENCY) + response = frequencyResponse (m_idFilter, m_bw, x, m_filterParam); return (response); } + +double +SignalFilter::spatialResponse (FilterID filterID, double bw, double x, double param, int nIntegral = 0) +{ + if (nIntegral == 0) + return spatialResponseAnalytic (filterID, bw, x, param); + else + return spatialResponseCalc (filterID, bw, x, param, nIntegral); +} + /* NAME * filter_spatial_response_calc Calculate filter by discrete inverse fourier * transform of filters's frequency @@ -232,9 +463,9 @@ SignalFilter::response (const char* filterName, const char* domainName, double b */ double -SignalFilter::spatialResponseCalc (double x, double param, int n) const +SignalFilter::spatialResponseCalc (double x, double param, int nIntegral) const { - return (spatialResponseCalc (m_idFilter, m_bw, x, param, n)); + return (spatialResponseCalc (m_idFilter, m_bw, x, param, nIntegral)); } double @@ -451,10 +682,9 @@ SignalFilter::spatialResponseAnalytic (FilterID filterID, double bw, double x, d double SignalFilter::integral_abscos (double u, double w) { - if (fabs (u) > F_EPSILON) - return (cos(u * w) - 1) / (u * u) + w / u * sin (u * w); - else - return (w * w / 2); + return (fabs (u) > F_EPSILON + ? (cos(u * w) - 1) / (u * u) + w / u * sin (u * w) + : (w * w / 2)); } @@ -519,3 +749,118 @@ for (int i = 0; i < np; i++) return (sum * dx); } + +void +SignalFilter::finiteFourierTransform (const float input[], complex output[], const int n, int direction) +{ + if (direction < 0) + direction = -1; + else + direction = 1; + + double angleIncrement = 2 * PI / n; + for (int i = 0; i < n; i++) { + double sumReal = 0; + double sumImag = 0; + for (int j = 0; j < n; j++) { + double angle = i * j * angleIncrement * direction; + sumReal += input[j] * cos(angle); + sumImag += input[j] * sin(angle); + } + if (direction < 0) { + sumReal /= n; + sumImag /= n; + } + output[i] = complex (sumReal, sumImag); + } +} + + +void +SignalFilter::finiteFourierTransform (const complex input[], complex output[], const int n, int direction) +{ + if (direction < 0) + direction = -1; + else + direction = 1; + + double angleIncrement = 2 * PI / n; + for (int i = 0; i < n; i++) { + complex sum (0,0); + for (int j = 0; j < n; j++) { + double angle = i * j * angleIncrement * direction; + complex exponentTerm (cos(angle), sin(angle)); + sum += input[j] * exponentTerm; + } + if (direction < 0) { + sum /= n; + } + output[i] = sum; + } +} + +void +SignalFilter::finiteFourierTransform (const float input[], complex output[], int direction) const +{ + if (direction < 0) + direction = -1; + else + direction = 1; + + for (int i = 0; i < m_nSignalPoints; i++) { + double sumReal = 0, sumImag = 0; + for (int j = 0; j < m_nSignalPoints; j++) { + int tableIndex = i * j; + if (direction > 0) { + sumReal += input[i] * m_vecFourierCosTable[tableIndex]; + sumImag += input[i] * m_vecFourierSinTable[tableIndex]; + } else { + sumReal += input[i] * m_vecFourierCosTable[tableIndex]; + sumImag -= input[i] * m_vecFourierSinTable[tableIndex]; + } + } + if (direction < 0) { + sumReal /= m_nSignalPoints; + sumImag /= m_nSignalPoints; + } + output[i] = complex (sumReal, sumImag); + } +} + +// (a+bi) * (c + di) = (ac - db) + (bc + da)i +#if 0 +void +SignalFilter::finiteFourierTransform (const complex input[], complex output[], int direction) const +{ + if (direction < 0) + direction = -1; + else + direction = 1; + + for (int i = 0; i < m_nSignalPoints; i++) { + double sumReal = 0, sumImag = 0; + for (int j = 0; j < m_nSignalPoints; j++) { + int tableIndex = i * j; + if (direction > 0) { + sumReal += input[i] * m_vecFourierCosTable[tableIndex]; + sumImag += input[i] * m_vecFourierSinTable[tableIndex]; + } else { + sumReal += input[i] * m_vecFourierCosTable[tableIndex]; + sumImag -= input[i] * m_vecFourierSinTable[tableIndex]; + } + } + if (direction > 0) { + sumReal /= m_nSignalPoints; + sumImag /= m_nSignalPoints; + } + output[i] = complex (sumReal, sumImag); + } +} +#endif + +void +SignalFilter::dotProduct (const double v1[], const complex v2[], complex output[], const int n) +{ + for (int i = 0; i < n; i++) + output[i] = v1[i] * v2[i]; +}