The DFT Filter Bank

To obtain insight into the operation of filter banks implemented using the FFT, this section will derive the details of the DFT Filter Bank. More general STFT filter banks are obtained by using different windows and hop sizes, but otherwise are no different from the basic DFT filter bank.

The Discrete Fourier Transform (DFT) is defined by [243]

where is the input signal at time , and . In this section, we will show how the DFT can be computed exactly from a bank of FIR bandpass filters, where each bandpass filter is implemented as a demodulator followed by a lowpass filter. We will then find that the inverse DFT is computed by remodulating and summing the output of this filter bank. In this way, the DFT filter bank is shown to be a perfect-reconstruction filter bank. The STFT is then an extension of the DFT filter bank to include non-rectangular analysis windows and a downsampling factor .

---

Subsections

• The Running-Sum Lowpass Filter
• Modulation by a Complex Sinusoid
• Making a Bandpass Filter from a Lowpass Filter
• Uniform Running-Sum Filter Banks
  • System Diagram of the Running-Sum Filter Bank
  • DFT Filter Bank
  • Inverse DFT and the DFT Filter Bank Sum

---

Order a Hardcopy of Spectral Audio Signal Processing

Previous: Computational Examples in Matlab
Next: The Running-Sum Lowpass Filter

---

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.