Flanging

• Flanging involves the summing together of a signal and a time-varying delayed version of itself.

• The input-output relationship for a flanger is given by:
  $$y[n] = x[n] + g x[n - M[n]],$$
  where $M[n]$ is the time-varying length of a delay line and $g$ is the “depth” of the flanging effect. A flanger block diagram is shown in Fig. 6.

  Figure 6: A digital flanger block diagram.

  

• Because the delay-line length, $M[n]$, must change continuously and smoothly through time, it is necessary to make use of an interpolating delay line.

• At any instant in time, the flanger is equivalent to a feedforward comb filter, which has a frequency response as shown in Fig. 7.

  Figure 7: Magnitude response of a feedforward comb filter with $M = 5$, $b_{0} = 1$, and $g = b_{M}$ = 0.1, 0.5, and 0.9.

  

• For $g > 0$, there are $M$ peaks in the frequency response, centered about the frequencies $\omega_{k} = 2 \pi k/M, k = 0, 1, \ldots, M-1$. Between these peaks, there are $M$ notches at intervals of $f_{s}/M$ Hz.

• As $M$ changes over time, the peaks and notches of the comb response are compressed and expanded. The spectrum of a sound passing through the flanger is thus accentuated and deaccentuated by frequency region in a time-varying manner.

• The delay-line length of a flanger is typically modulated by a low-frequency oscillator (LFO). Oscillator waveforms are typically sinusoidal, triangular, or exponential.

• For a sinusoidally varied delay,
  $$M[n] = M_{0} \cdot (1 + A \sin[2 \pi f n T_s]),$$
  where $f$ is the flanger “rate” in Hz, $A$ is the “excursion” (maximum delay swing), $M_{0}$ is the average delay-line length that controls the average notch density, and $T_s$ is the sample period.

• For values of $-1 \leq g \leq 0$, the peaks and notches of the comb filter trade places. In practice, $g$ is normally contrained to the interval $[0,1]$ and the option of sign inversion is provided by a “phase inversion” switch.

• In the inverted mode, a notch is located at zero frequency. As a result, bass response will likely be weakened.

• Some flangers also implement feedback (in addition to the delayed feedforward path), which introduces spectral peaks and notches as previously described for feedback comb filters.

• The following MSP patch implements a flanger.