ios - 在 Accelerate vDSP 上进行离散傅里叶变换后进行逆离散傅里叶变换不会产生原始值

Question

我有一系列离散值。所以，我没有输入部分的虚数值。我正在对这些值进行离散傅里叶变换并执行离散傅里叶逆变换，以获取相同的值。这个想法是为了测试来自 Accelerate Framework 的 vDSP 是否按预期工作，因为文档是一堆废话，你必须自己弄清楚。

考虑以下代码

  lazy var DFTSetupForward: vDSP_DFT_Setup = {
    guard let setup = vDSP_DFT_zop_CreateSetupD(
      nil,
      vDSP_Length(self.numSamples),
      vDSP_DFT_Direction.FORWARD) else {
        fatalError("can't create vDSP_DFT_Setup")
    }
    return setup
  }()

  lazy var DFTSetupInverse: vDSP_DFT_Setup = {
    guard let setup = vDSP_DFT_zop_CreateSetupD(
      nil,
      vDSP_Length(self.numSamples),
      vDSP_DFT_Direction.INVERSE) else {
        fatalError("can't create vDSP_DFT_Setup")
    }
    return setup
  }()
 func discreteFourierTransform (_ valores:[Double] = []) -> ([Double], [Double]) {
    let numeroDados = valores.count

    let inputImag  = Array<Double>(repeating:0.0, count:numeroDados)
    var outputReal = Array<Double>(repeating:0.0, count:numeroDados)
    var outputImag = Array<Double>(repeating:0.0, count:numeroDados)

    vDSP_DFT_ExecuteD(DFTSetupForward, valores, inputImag, &outputReal, &outputImag)
    return (outputReal, outputImag)
  }


  // faz a DISCRETE FOURIER TRANSFORM DOUBLE
  // a saída corresponde aos vetores reais e imaginários
  func discreteFourierTransformInverse (_ valoresReais:[Double] = [],
                                        _ valoresImaginarios:[Double] = []) -> ([Double], [Double]) {
    let numeroDados = valoresReais.count

    var outputReal = Array<Double>(repeating:0.0, count:numeroDados)
    var outputImag = Array<Double>(repeating:0.0, count:numeroDados)

    vDSP_DFT_ExecuteD(DFTSetupInverse, valoresReais, valoresImaginarios, &outputReal, &outputImag)
    return (outputReal, outputImag)
  }

和后来的这些电话

let (outputDFTreal, outputDFTImaginario ) = self.discreteFourierTransform(normalizedData)

let (sinalRealX, sinalImaginarioX ) = self.discreteFourierTransformInverse(outputDFTreal, outputDFTImaginario)

或者换句话说，我从第一个 let 中获取傅里叶变换的结果，并将它们注入(inject)到傅里叶逆变换中。我希望 sinalRealX 等于原始数据，在这种情况下是 normalizedData 并且还希望 signalImaginarioX 全部为零。

但值(value)观完全不同!

这是怎么回事？

Answer 1

您不应期望 IDFT 返回标准化值。您应该期望它返回原始(预归一化)值，我相信它会返回(在计算错误范围内):

let fft = FFT()
let data = Array(stride(from: 0.0, to: Double(fft.numSamples), by: 1))
let normalizedData = data.map{ $0/Double(fft.numSamples) }
let (outputDFTreal, outputDFTImaginario ) = fft.discreteFourierTransform(normalizedData)
let (sinalRealX, sinalImaginarioX ) = fft.discreteFourierTransformInverse(outputDFTreal, outputDFTImaginario)
print(data)
print(sinalRealX)
print(sinalImaginarioX)

let zeros = Array(repeating: 0.0, count: data.count)
func avgerr(_ a: [Double], _ b: [Double]) -> Double {
    return zip(a, b).map { abs($0 - $1) }.reduce(0, +) / Double(a.count)
}

avgerr(data, sinalRealX)  // 2.442490654175344e-15
avgerr(zeros, sinalImaginarioX) // 2.602442788260593e-15