Spectral Interpolation

The need for spectral interpolation comes up in many situations. For example, we always use the DFT in practice, while conceptually we often prefer the DTFT. For time-limited signals, that is, signals which are zero outside some finite range, the DTFT can be computed from the DFT via spectral interpolation.^5.1 Another application of DFT interpolation is spectral peak estimation; in this situation, we obtain a sampled spectral peak from a DFT, and interpolation is used to estimate the frequency of the peak more accurately than what is obtained by rounding to the nearest DFT bin frequency.

In this and the following section, we will discuss two types of spectral interpolation:

• Ideal spectral interpolation (zero-padding in the time domain)
• Parabolic interpolation (fitting a parabola at the sampled peak)

When these methods are used together, we have what we call the quadratically interpolated FFT (QIFFT) method [248,1]. The QIFFT method can be considered an approximate maximum likelihood method for spectral peak estimation, as we will see.

---

Subsections

• Ideal Spectral Interpolation
• Interpolating a DFT
• Zero Padding in the Time Domain
  • Practical Zero Padding
    • Zero-Padding to the Next Higher Power of 2
    • Zero-Padding for Interpolating Spectral Displays
    • Zero-Padding for Interpolating Spectral Peaks
  
  
• Zero-Phase Zero Padding
  • Matlab/Octave fftshift utility
    • Index Ranges for Zero-Phase Zero-Padding
    • Summary
  
  
• Example: Spectrum Analysis of an Oboe Tone
• Quadratic Peak Interpolation
  • Quadratic Interpolation of Spectral Peaks
  • Phase Interpolation at a Peak
  • Matlab for Parabolic Peak Interpolation
  
  
• Bias of Parabolic Peak Interpolation
• Optimal Peak-Finding in the Spectrum
  • Minimum Zero-Padding for High-Frequency Peaks
  • Minimum Zero-Padding for Low-Frequency Peaks
  • Matlab for Computing Minimum Zero-Padding Factors
  • Least Squares Sinusoidal Parameter Estimation
    • Sinusoidal Amplitude Estimation
    • Sinusoidal Amplitude and Phase Estimation
    • Sinusoidal Frequency Estimation
  • Maximum Likelihood Sinusoid Estimation
  • Likelihood Function
    • Multiple Sinusoids in Additive Gaussian White Noise
    • Non-White Noise
  • Generality of Maximum Likelihood Least Squares
  
  
• Problems
• Lab Assignments

---

Order a Hardcopy of Spectral Audio Signal Processing

Previous: Conclusion
Next: Ideal Spectral Interpolation

---

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.