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I have developed a spectrum scope that plots signal vs. frequency starting from IQ signals. I currently do not have a way to balance the IQ signals, so am looking at various ways to do so. I have come across this article, which is reasonably easy for me to follow since I am fairly new to all DSP. The one question I have is concerning the notation of some of the equations: For example, $a^2 = 2\langle t(i) * t(i)\rangle$. What does $\langle\cdot \rangle$ represent? I can find no reference for these in mathematical symbol tables.

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    $\begingroup$ The paper provides a definition: the average of the signal computed over N periods. $\endgroup$ – pichenettes Nov 1 '14 at 22:29
  • $\begingroup$ Hi Thanks, yes my math is quite rusty! Can anyone explain how many samples would provide a good value? $\endgroup$ – Tom Nov 3 '14 at 22:58
  • $\begingroup$ According to this page, that symbol means "inner product". See this answer to this question. $\endgroup$ – jeremy Oct 5 '16 at 19:20
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I wrote code in Java based on that article but it isn't working for me. Alpha (I-data scalar) calculates to be around $100$ and $cos(\omega)$ often calculates to an imaginary number. Did anyone have any luck with this or could point me to a better IQ Correction algorithm?

(To be clear: calculateMatrix, a, c, and d are all instance variables)

public void correctIQ(int[] data) {

    int length = data.length / 2;

    // ( I'(t) & Q'(t) )
    int[] iData = new int[length];
    int[] qData = new int[length];

    // Deinterleave IQ data 
    for (int i = 0; i < length; i++) {
        iData[i] = data[i * 2];
        qData[i] = data[i * 2 + 1];
    }

    // ( I''(t) & Q''(t) )
    double[] qDataOffset = new double[length];
    double[] iDataOffset = new double[length];

    // Calculate DC offset
    double qOffset = mean(qData);
    double iOffset = mean(iData);

    // Apply DC offset
    for (int i = 0; i < length; i++) {
        iDataOffset[i] = iData[i] - iOffset;
        qDataOffset[i] = qData[i] - qOffset;
    }

    // Provided equations calculated on frequency change
    if(calculateMatrix) {
        double alpha = Math.sqrt(2.0 * multiplyMean(iDataOffset, iDataOffset));
        double sinw = (2.0 / alpha) * multiplyMean(iDataOffset, qDataOffset);
        double cosw = Math.sqrt(1.0 - (sinw * sinw));

        a = 1.0f / alpha;
        c = -sinw / (alpha * cosw);
        d = 1.0f / cosw;
    }        

    // Replace into interleaved array
    for(int i=0;i<length;i++) {
        data[i*2]=(int)((iData[i]-iOffset)*a);
        data[i*2+1]=(int)((iData[i]-iOffset)*c+(qData[i]-qOffset)*d);
    }
}

double mean(int[] values) {
    long sum = 0L;
    for(int i=0;i<values.length;i++) {
        sum+=values[i];
    }
    return ((double)sum)/values.length;
}

double multiplyMean(double[] values1, double[] values2) {
    long sum = 0L;
    for(int i=0;i<values1.length;i++) {
        sum+=values1[i]*values2[i];
    }
    return ((double)sum)/values1.length;
}
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I didn't get a chance to look through the article in detail, but it looks like taking the inner product between 2 vectors over some index variable (in your case, $i$), as in $<X_i, X_i> = \frac{1}{N}\sum_i |X_i|^2$.

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This is the averaging symbol and you see it quite often in DSP and communication systems books and articles. It's right there in article also:

enter image description here

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