Recipe 22.1. Using a Window’s DTFT Response to Assess the Frequency Response for a Single Bin in a Windowed DFT

This analysis determines the response of a single DFT bin, say, bin k, to signals over the entire range of frequencies from –(2T)^–1 to (2T)^–1, where T is the time-domain sampling interval.

1. Center a copy of the window’s DTFT response, W(f), over the frequency corresponding to bin k. This frequency-shifted copy of the response can be denoted as W_k(f) = W(f – kF).
2. For an input signal consisting of an ideal sinusoid of cyclic frequency f_c, the DFT bin k response due to the positive-frequency portion of the input signal’s spectrum is determined as

   

3. The DFT bin k response due to the negative-frequency portion of the input signal’s spectrum is determined as

   

4. The total bin k response is then determined as

   X[k] = X⁺[k] + X^–[k]

Example 22.4. Analyzing Triangular Window Performance Using a Bin-centric Strategy

Assume that a triangular window is applied to the input of a 32-point DFT. Use Recipe 22.1 to determine the response in a single DFT bin, say, bin 6, when the input signal is

where F is the frequency interval between DFT bins. Figure 22.1 shows the real and imaginary components for W₆(f). As indicated in the figure, the DFT bin 6 response due to the positive-frequency portion of the input spectrum is obtained as

X⁺[6] = –0.149672 – j0.201809

and the bin 6 response due to the negative-frequency portion of the input spectrum is obtained as

X^–[6] = –0.0298469 –j0.040244

The total bin 6 response is then

X[6] = X⁺[k] + X^–[^k]

                       =–0.179519 – j0.161565

The magnitude of the response is

Figure 22.1. DTFT response for a triangular window after response has been shifted to align the center of the main lobe with the frequency corresponding to DFT bin 6: (a) real part, (b) imaginary part

Example 22.5. Analyzing Triangular Window Performance Using a Signal-centric Strategy

Assume that a triangular window is applied to the input of a 32-point DFT. Use Recipe 22.2 to determine the response in the DFT bins when the input signal is

where F is the frequency interval between DFT bins. The real and imaginary parts of the composite continuous-frequency response, X(f), obtained using Eq. (R22.1) from Recipe 22.2, are shown in Figure 22.3. The magnitude of this response is plotted with the corresponding DFT magnitude response, |X [m]|, in Figure 22.4. Notice that the bin 6 response matches the response obtained using the bin-centric approach in Example 22.1.

Figure 22.2. Analysis strategy for signal-centric approach. One copy of the DTFT spectrum (solid trace) is centered at frequency f_c, and a second copy (dotted trace) is centered at frequency NF – f_c

Figure 22.3. DTFT response for a triangular-windowed input signal x[n] = cos(2π f_cnT): (a) real part, (b) imaginary part

Figure 22.4. DTFT magnitude response (solid trace) and DFT magnitude response (circles) for a triangular windowed, 32-sample segment of input signal x[n] = cos(2π f_cnT)

Recipe 22.2. Signal-Centric Approach for Assessing the Response of a Windowed DFT

This analysis determines the response of every DFT bin to a signal component comprising a single real sinusoid at frequency f_c.

1. Center a copy of the window’s DTFT response, W (f), at frequency f_c and a second copy at NF – f_c, as indicated in Figure 22.2. These frequency-shifted copies of the response can be denoted as W (f – f_c) and W (f – NF + f_c), respectively. It would be cumbersome to plot two copies of a complex DTFT response in a single figure, therefore Figure 22.2 shows the magnitude responses |W (f – f_c)| and |W (f – NF + f_c)| just to convey a sense of the frequency relationship between the two copies of the response.
2. The composite continuous-frequency response, X(f), is obtained as the appropriately phased complex sum of W (f – f_c) and W (f – NF + f_c)

   R22.1

   

   The DFT response, X[m], is obtained by sampling X(f) at the DFT bin frequencies