Circuits and Systems, 2013, 4, 117-122

http://dx.doi.org/10.4236/cs.2013.41017 Published Online January 2013 (http://www.scirp.org/journal/cs)

Performance Analysis of an Inverse Notch Filter and Its

Application to F0 Estimation

Yosuke Sugiura, Arata Kawamura, Youji Iiguni

Graduate School of Engineering Science, Osaka University, Osaka, Japan

Email: yosuke@sip.sys.es.osaka-u.ac.jp

Received November 21, 2012; revised December 21, 2012; accepted December 29, 2012

ABSTRACT

In this paper, we analyze an inverse notch filter and present its application to F0 (fundamental frequency) estimation.

The inverse notch filter is a narrow band pass filter and it has an infinite impulse response. We derive the explicit forms

for the impulse response and the sum of squared impulse response. Based on the analysis result, we derive a normalized

inverse notch filter whose pass band area is identical to unit. As an application of the normalized inverse notch filter, we

propose an F0 estimation method for a musical sound. The F0 estimation method is achieved by connecting the normal-

ized inverse notch filters in parallel. Estimation results show that the proposed F0 estimation method effectively detects

F0s for piano sounds in a mid-range.

Keywords: Band Pass Filter; Inverse Notch Filter; Impulse Response Analysis

1. Introduction

In speech processing, image processing, biomedical sig-

nal processing, and many other signal processing fields,

it is important to eliminate the narrowband signal. The

examples of the narrowband signal are a hum noise from

the power supply, an acoustic feedback, and an interfere-

ence noise, and so on. A notch filter is useful for the

elimination of the narrowband signal [1-7], where the

notch filter passes all frequencies expect of a stop fre-

quency band centered on a center frequency, called as the

notch frequency. The notch filter has a simple structure,

and its stop bandwidth and its notch frequency are indi-

vidually designed. The notch filter is used in many ap-

plications and it has been analyzed in many literatures

[1,4-7].

On the other hand, an inverse notch filter is a band

pass filter which has the inverse characteristics of the

notch filter. In contrast to the notch filter, there are few

applications of the inverse notch filter. As an example of

the applications, an active noise control system for re-

ducing a sinusoidal noise has been proposed [8]. In this

system, the inverse notch filter is used to extract the si-

nusoidal noise. Unfortunately, the system is designed

without respect to the impulse response of the inverse

notch filter. Hence, the inverse notch filter cannot accu-

rately extract the sinusoidal noise when the filter output

is in the transient state. To utilize the inverse notch filter

more effectively for not only the active noise control

system but also many other applications, a more detail

analysis of the impulse response for the inverse notch

filter needs to be required.

In this paper, we derive an explicit form for the infinite

impulse response of the inverse notch filter. Additionally,

we derive an explicit form for the sum of the squared

impulse response. Then, we reveal the limit values of

these two infinite sequences. Next, based on the analysis

results, we propose a normalized notch filter whose pass

band area is adjusted to unit. The normalized inverse

notch filter is efficient to estimate the output power in the

short time such as the frame processing. Finally, as an

application of the normalized inverse notch filter, we

present an F0 estimation method for a musical sound. In

the F0 estimation method, we use multiple normalized

inverse notch filters whose pass frequencies are identical

to F0s for each monophonic sound, respectively. These

normalized inverse notch filters are connected in parallel.

In the estimation procedure, we detect F0 from the in-

verse notch filter whose output power is largest among

all the inverse notch filter output powers. From the

simulation results, we see that the proposed F0 estimation

method can effectively detect the F0 both of for the

monophonic sounds and the polyphonic sounds.

2. Performance Analysis of Inverse Notch

Filter

In this section, we explain both of the notch filter and the

inverse notch filter, where the latter filter has an inverse

characteristic of the notch filter. The notch filter passes

all frequencies expect of the narrow frequency band cen-

tered on the notch frequency. The stop bandwidth and the

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