NOISE REDUCTION IN SPEECH ENHANCEMENT BY SPECTRAL SUBTRACTION WITH SCALAR KALMAN FILTER

Sinusoidal signal have a very important property that no other periodic signals have, that is the sum of two sinusoidal signals with same frequency is another sinusoidal signal with same

VIETNAM NATIONAL UNIVERSITY, HANOI UNIVERSITY OF ENGINEERING AND TECHNOLOGY

VIETNAM NATIONAL UNIVERSITY, HANOI UNIVERSITY OF ENGINEERING AND TECHNOLOGY

Đặng Minh Công

NOISE REDUCTION IN SPEECH ENHANCEMENT BY SPECTRAL SUBTRACTION WITH SCALAR KALMAN

FILTER

Major:Computer Science

Supervisor:Assoc Prof Dr Nguyễn Đình Việt

AUTHORSHIP

“I hereby declare that the work contained in this thesis is of my own and has not been previously submitted for a degree or diploma at this or any other higher education institution To the best of my knowledge and belief, the thesis contains no materials previously published or written by another person except where due reference or acknowledgement is made.”

Signature:………

SUPERVISOR’S APPROVAL

“I hereby approve that the thesis in its current form is ready for committee examination as a requirement for the Bachelor of Computer Science degree at the University of Engineering and Technology.”

Signature:………

I studied at UET

ABSTRACT

In the system that related to speech communication like telecommunication system or speech processing, the presence of background noise in speech signal is undesirable Background noise can make the user harder to hear the speech, or decrease the performance of speech processing systems Therefore, to enhance the quality of speech signal, noise reduction is an important problem

In this thesis, we present a single channel noise reduction method for speech enhancement This method is based on the principle of spectral subtraction methods, with the addition of using scalar Kalman Filter for residual noise removal It models the changing of speech magnitude spectrum as Gaussian random process and the magnitude residual noise as Gaussian white noise for applying scalar Kalman Filter The scalar Kalman Filter used in this method is designed in order to be suitable for the characteristics of speech and noise signal Our obtained experiment results with the online NOIZEUS speech corpus show that the presented method has consistent improved the SNR measures of noisy speech signal In overall, experiment results also show that the SNR improvement

of the presented method is better than other basic implementations of spectral subtraction

TÓM TẮT

Trong các hệ thống liên quan đến truyền thông bằng tiếng nói con người như hệ thống viễn thông hoặc xử lý tiếng nói, sự hiện diện của nhiễu trong tiếng nói là không mong muốn Tiếng ồn xung quanh được thu cùng với tiếng nói có thể làm cho người dùng khó khăn hơn để nghe các bài phát biểu, hoặc làm giảm hiệu suất của hệ thống xử lý tiếng nói Vì vậy, để nâng cao chất lượng của tín hiệu tiếng nói, giảm nhiễu là một vấn đề quan trọng

Trong khóa luận này, chúng tôi trình bày một phương pháp giảm nhiễu để nâng cao chất lượng tiếng nói Phương pháp này dựa trên nguyên tắc của phương pháp trừ phổ, bổ sung thêm việc sử dụng bộ lọc Kalman một chiều để loại bỏ nhiễu tàn

dư Phương pháp này mô hình hóa sự thay đổi của phổ biên độ giọng nói theo thời gian như quá trình ngẫu nhiên Gauss và phổ biên độ nhiễu tàn dư là nhiễu trắng Gauss để áp dụng bộ lọc Kalman một chiều Bộ lọc Kalman sử dụng trong phương pháp này được thiết kế để phù hợp với đặc điểm của tín hiệu giọng nói và nhiễu

Kết quả thử nghiệm của chúng tôi với bộ dữ liệu mẫu tiếng nói NOIZEUS trực tuyến cho thấy rằng phương pháp trình bày đã cải thiện được số đo SNR của tín hiệu tiếng nói bị nhiễu Nhìn chung, kết quả thử nghiệm cũng cho thấy sự cải thiện SNR của phương pháp trình bày là tốt hơn so với các cài đặt cơ bản khác

của phép trừ phổ

TABLE OF CONTENTS

List of Figures 10

List of Tables 11

ABBREVATIONS 12

Chapter 1 INTRODUCTION 13

1.1 Motivation 13

1.2 Survey of existing methods 13

1.3 Contributions 14

1.4 Structure of the Thesis 14

Chapter 2 BACKGROUND 15

2.1 Sound 15

2.2 Human perception of sound 17

2.2.1 Loudness 17

2.2.2 Pitch 18

2.2.3 Timbre 18

2.3 Audio Signal 19

2.3.1 Analog audio signal 19

2.3.2 Digital audio signal 20

2.3.3 Sampling 20

2.3.4 Quantization 22

2.4 Fourier Transform and Frequency domain representation 22

2.5 Kalman Filter 25

Chapter 3 NOISE REDUCTION BY SPECTRAL SUBTRACTION WITH SCALAR KALMAN FILTER 26

3.1 Spectral Subtraction 26

3.1.1 Principle 26

3.1.2 Half-wave Rectification 28

3.1.3 Residual noise 28

3.2.3 Measurement noise variance R 49

3.2.4 Process noise variance Q 32

3.2.5 Algorithm 33

Chapter 4 EVALUATION 54

4.1 Objective Measures of Speech Quality 54

4.1.1 SNR 54

4.1.2 Segmental SNR (SNRseg) 55

4.2 Experiment setup 35

4.3 Experiment results 57

Chapter 5 CONCLUSION 43

5.1 Conclusions 43

5.2 Future Works 43

Bibliography 45

Appendix A MATLAB source code of the implementation 47

List of Figures

Figure 1: Sound signals of some musical instruments 15

Figure 2: Musical notes in a piano keyboard 18

Figure 3: Waveform of two particular signals with the same sinusoidal components combined in a different ways 19

Figure 4: Sampling of a sinusoidal analog signal 20

Figure 5: Sampling process with low sampling rate 21

Figure 6: Block diagram of spectral subtraction 29

Figure 7: Flowchart of Kalman Filter with each frequency component 52

Figure 8: Block diagram of the presented method 52

Figure 9: SNR and SNRseg results of three methods with sp07_car_sn0.wav 40

Figure 10: Waveform of the clean speech signal sp07.wav 41

Figure 11: Waveform of noisy speech sp07_car_sn0.wav after noise reduction by proposed method 41

Figure 12: Waveform of noisy speech sp07_car_sn0.wav after noise reduction by Boll spectral subtraction 41

Figure 13: Waveform of noisy speech sp07_car_sn0.wav after noise reduction by Berouti spectral subtraction 41

List of Tables

Table 1: Different kinds of signal and their Fourier Transforms 24

Table 2: Experiment results with the speeches corrupted by car noise at SNR 0dB 58

Table 3: Experiment results with the speeches corrupted by car noise at SNR 5dB 37

Table 4: Experiment results with the speeches corrupted by car noise at SNR 10dB 38

Table 5: Experiment results with the speeches corrupted by car noise at SNR 15dB 39

Table 6: Average SNR and SNRseg gain when compare three methods’ results with noisy speech 39

Table 7: Improvements of proposed method compared to other two methods 40

ABBREVATIONS

Chapter 1 INTRODUCTION

1.1 Motivation

Although there are many methods of communication in our days, speech-based communication is still the primary mode of interaction between humans Not only that, there are many real life situations that automatic speech recognition and voice user interface are having practical uses Therefore, research of speech-related problems in Computer Science will continue for a long time into future

There is a very common problem encountered in speech-related systems, which is noise In the presence of acoustic noise, the quality of speech signal is degraded and the level of degradation will depend on characteristics of noise and environment In telecommunication, the presence of noise will make the users harder to hear the speech, or in speech recognition system, the performance of the system will be affected by noise Hence, it is essential to devise algorithms for automatic noise reduction in speech-related systems

1.2 Survey of existing methods

Noise reduction in speech enhancement is well-researched problem and there have been many proposed methods for this problem In general, there are two classes of noise reduction algorithms: Single-channel noise reduction (only one microphone is used) and Multiple-channel noise reduction (multiple microphones are used)

Single-channel methods utilise the temporal and/or spectral differences between the speech and noise signals to suppress the noise While only need one microphone to work, single-channel methods sometimes cannot reduce the noise effectively enough, especially in the case that noise and speech signals are overlapped in time-frequency domain On the other hand, multiple-channel methods, which utilise temporal, spectral and also spatial differences, are capable of more effective noise reduction than single-channel methods, but with the cost

of using more microphones Because multiple-channel methods are not the focus of this thesis, we will not mention them further on

In single-channel noise reduction, spectral subtraction is one of the most popular classes of methods It bases on the principle that we can get an estimate of original speech by subtracting the spectrum of noise from the spectrum of noisy signal It also belongs to the earliest single-channel methods, with the first paper published in 1979 [1] While being simple to implement and effective for reducing noise in low level noise cases, spectral subtraction methods’ performance is worsen off in high noise level cases And spectral subtraction methods also introduce a new kind of artefacts, which was called as musical noise, into the estimated signal

Another class of methods is using Kalman filter to reduce the noise in speech [2] Kalman filter operates recursively on a stream of past input noisy data and its output is a statistical estimate of current observed data By modelling the speech signal as AR process and applying Kalman filter, significant noise reduction can be achieved when linear prediction coefficients are estimated from clean speech signal However, when the parameters cannot be estimated correctly enough, the Kalman filter’s performance is suffered [3]

1.3 Contributions

The main contribution of this thesis can be summarised as follow:

 Modification of the spectral subtraction method proposed Boll [1] The modification

is using the Kalman Filter for residual noise removal

 Experimental evaluation of the effectiveness of the aforementioned modification

1.4 Structure of the Thesis

The remaining parts of this thesis are organized as follow: Chapter 2 presents the background knowledge which this thesis based on Chapter 3 describes the spectral subtraction method and my modification In Chapter 4, we show and discuss the experimental results Finally, in Chapter 5, we summarise the main contributions of this thesis and possible future works

Chapter 2 BACKGROUND

𝑥(𝑡) = 𝑓(𝑡) (𝑥 𝑖𝑠 𝑎 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑠𝑜𝑢𝑛𝑑 𝑚𝑒𝑑𝑖𝑢𝑚, 𝑓(𝑡) 𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡𝑖𝑚𝑒)

Sound detector is essentially a device that measures sound signal Microphone, human and animal ears are examples of sound detectors

Sound has many properties and one of the most important properties is frequency With a periodic signal, frequency is the number of times that it repeats during a unit of time For example, a frequency of 1 Hz means that the signal repeats itself one time in one second Strictly periodic sound signal (repeat itself exactly) is impossible to produce in reality, but quasi-periodic sound signal (repeat not exactly) is possible Figure 1 below is an example of some quasi-periodic signals produced by musical instruments in reality:

Figure 1: Sound signals of some musical instruments

An important kind of periodic signal is sinusoidal signal:

𝑥(𝑡) = 𝐴 sin(2𝜋𝑓𝑡 + 𝜑)

𝐴 is the peak amplitude of signal, 𝑓 is the frequency of signal, and 𝜑 is called the phase Sinusoidal signal have a very important property that no other periodic signals have, that is the sum of two sinusoidal signals with same frequency is another sinusoidal signal with same frequency:

𝐴1𝑠𝑖𝑛(2𝜋𝑓𝑡 + 𝜑1) + 𝐴2𝑠𝑖𝑛(2𝜋𝑓𝑡 + 𝜑2) = 𝐴3𝑠𝑖𝑛(2𝜋𝑓𝑡 + 𝜑3) The French mathematician Fourier (1768-1830) was the first one who pointed out that any periodic function can be represented as an infinite sum of sines and cosines functions [4] That means any periodic signal can be analysed into the sum of many sinusoidal signals A periodic signal with frequency 𝑓 will be represented as:

It is natural to speak about the frequency of periodic signal, but it turns out that aperiodic signal also has “frequency” too The mathematicians find out that even aperiodic signal can

be expressed in terms of sinusoidal signals The discrete sum in the case of periodic signal becomes the continuous integral in the case of aperiodic signal:

This theory is named as Fourier analysis, in honour of Joseph Fourier, who was the first person proposed this theory Since then, it has important roles in many scientific fields, from mathematics, physics, acoustics, signal processing, etc

Frequency of sound signal has an important role in how human perceive sound, which we will mention in the next section

2.2 Human perception of sound

It is common fact that human cannot hear some sounds that are audible to some kinds of animal Even human adults cannot hear the sounds that the children can hear The scientists discovered that normal people can only hear the sounds which have the frequency ranging from 20 Hz to 20 kHz It does not mean that human can only hear periodic sounds with frequency in that range It means that when a sound signal is decomposed into many sinusoidal components, only the components with frequency falling in that range is perceived

by human So, even a periodic sound signal with frequency in 20 Hz to 20 kHz range cannot

be perceived fully by human if it has sinusoidal components at frequency outside of that range

Human perception of sound can be divided into three primary characteristics: Loudness, Pitch and Timbre

2.2.1 Loudness

Human perception of loudness is mostly depended on the amplitude of sound signal With the sinusoidal sound signal:

𝑥(𝑡) = 𝐴 sin(2𝜋𝑓𝑡 + 𝜑) The higher the value of 𝐴, the louder the sound is However, the relation between amplitude and loudness is not linear That means when the value of 𝐴 increases 𝑛 times, the perceived loudness does not increase at the same rate In fact the relation between them is logarithmic

In practice, we use the Sound Pressure Level (SPL) measure instead of amplitude when talking about the loudness of sound (the name pressure is used because sound signal is commonly taken to be the variation of medium’s pressure):

𝐿𝑝 = 20 log10(𝑝

𝑝0) 𝑑𝐵

𝑝 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑜𝑜𝑡 𝑚𝑒𝑎𝑛 𝑠𝑞𝑢𝑎𝑟𝑒 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 (𝑠𝑜𝑢𝑛𝑑 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒)

𝑝0 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑠𝑜𝑢𝑛𝑑 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 The common chosen sound pressure reference is 𝑝0 = 20 𝜇𝑃𝑎, which is the lowest sound pressure human can detect (Pa, or Pascal, is the SI unit of pressure) So the lowest level is defined to be 0 𝑑𝐵, but the highest level is not clearly defined Normally, the threshold of feeling is taken to be 120 𝑑𝐵 and the threshold of pain is taken to be 140 𝑑𝐵 Listeners can detect a change in loudness when 𝐿𝑝 is increased by 1 𝑑𝐵 (which means the amplitude is multiplied by 101 20 ⁄ ≈ 1.12 times) Acousticians tell us that an increase of 10 𝑑𝐵 (101 2 ⁄ ≈3.16 times of current amplitude) will give us an impression of double the loudness

This non-linearity gives human ear an interesting characteristic that it is sensitive to a very small change in amplitude at low sound pressure level, but is very insensitive to amplitude change at high sound pressure level

Aside from amplitude, frequency also has an impact on the perception of loudness While the human hearing range is from 20 Hz to 20 kHz, human ear is far more sensitive to the range of

1 kHz to 4 kHz For example, listeners can detect the 0 𝑑𝐵 𝑆𝑃𝐿 sounds at 3 kHz, while the sounds at 100 Hz require the sound pressure level of 40 𝑑𝐵 in order to be audible

2.2.2 Pitch

Pitch is what gives us the impression of “higher” or “lower” sounds For example, we have a feeling that female voices are higher than male voices, therefore, we say that the female voices have higher pitch than the male voices

The perception of pitch is directly related to the fundamental frequency of a sound signal, so that we commonly use frequency to measure pitch However, we must take note that pitch is

a perception, while frequency is a property of objectively existing sound, not depended on human sensation

Pitch is directly related to the fundamental frequency, so aperiodic signals without fundamental frequency do not make any particular pitch Most musical instruments produce periodic sounds; only some of them produce aperiodic sounds, like drum or cymbal Human ear finds the sounds with fundamental frequency pleasing, while the sounds without any particular pitch annoying

Similar to loudness, human perception of pitch is non-linear and logarithmic Musical pitches are organized into octaves The increment of one octave means double the fundamental frequency But every octave only consists of 12 different musical notes That means with the low pitch sounds, human ear is more sensitive to the change in frequency than with the high pitch sounds The below figure of piano keyboard from [5] clarifies that fact:

Figure 2: Musical notes in a piano keyboard 2.2.3 Timbre

Timbre is a complicated perception With a same musical note, timbre is what helps us know

it is produced by piano, guitar, violin, or human singer People often say that timbre is

Figure 3: Waveform of two particular signals with the same sinusoidal components

combined in a different ways

We can see that the waveforms of both signals are different, but human ear perceives them as the same sound This is an interesting fact Let examined the two above signals more:

𝑥1(𝑡) = 𝐴1sin(2𝜋1000𝑡) + 𝐴2sin(2𝜋3000𝑡)

𝑥2(𝑡) = 𝐴1sin(2𝜋1000𝑡) − 𝐴2sin(2𝜋3000𝑡) = 𝐴1sin(2𝜋1000𝑡) + 𝐴2sin(2𝜋3000𝑡 + 𝜋)

It turns out that the only difference between two signals is the phase of 3 𝑘𝐻𝑧 sinusoidal component The key of problem is the fact that human ear is sensitive to the amplitude of each sinusoidal components but very insensitive to the phase of them

Why the ear is insensitive to phase information? It is because of the fact that sound propagations of different frequencies are different, so that sounds of different frequencies will reach the ear through different paths Therefore, when we change our positions, the phase of different sinusoidal components will change by different amount, while their amplitude changes are relatively similar to each other If human ear is sensitive to phase information, then we will feel that the sounds are drastically depended on hearing position, even when the sound sources remain unchanged This is undesirable, and the natural evolution had proven it

by the phase insensitivity of human ear

2.3 Audio Signal

2.3.1 Analog audio signal

Before the advent of audio signal, manipulation of sound is not an easy task It means the manipulation of the medium of sound In the late 19th century, Thomas Edison was the first person who converted sound signal into electrical signal and converted back the electrical signal into sound That converted signal from sound is called audio signal With audio signal,

if we want to manipulate a sound signal, we just need to convert it into electrical audio signal,

let it run through a circuit and convert it back into sound The audio signal processing field was born with the advent of audio signal

The first kind of audio signal was called analog audio signal, because it was analogue to sound signal Today, when we talk about analog signal, it often means the signal is continuous, in contrast with digital signal

2.3.2 Digital audio signal

Since the advent of digital computer, digital audio signal has gradually replaced the place of analog audio signal, and nowadays, it is the most common form of audio signal While the range and domain of analog signal are continuous, the range and domain of digital signal are discrete While the usage of digital signal lead to inevitable information loss from analog signal, it has the advantage that we can apply exact manipulation on them Analog signal on the other hand, is harder to manipulate because of physical constraints and noises during processing and transmission

The converting process from analog signal to digital signal is called sampling and quantization

2.3.3 Sampling

Sampling is the process that converts the continuous domain of analog signal into a discrete domain It means that a continuous signal is converted into a discrete series of values The ratio of conversion is called sampling rate If a one second analog signal is converted into a series of 100 values, then the sampling rate is 100 Hz (it has the same unit as frequency) For example, Figure 4 below shows a sine wave signal and the result signal after sampling:

The above case is an example of proper sampling Because there is only one sine wave (with frequency not lower than the original signal) that can match with the sampled signal, therefore we can reconstruct the original analog signal from the sampled signal However, not all sampling processes are proper For example:

Figure 5: Sampling process with low sampling rate

In this case, the sampling rate is lower than the frequency of sinusoidal signal As we can see from the Figure 5, there is another sinusoidal signal that matches with the sampled signal Therefore, we cannot restore the original signal from the sampled signal As a general rule, the higher the sampling rate, the better But depend on the actual application; we only need a particular high sampling rate For example, the sampling rate in Figure 4 is high enough; we

do not need a higher sampling rate

The Nyquist-Shannon sampling theorem indicates that if a signal x(t) is sampled at regular intervals of time and at a rate higher than twice the highest signal frequency, then the samples contain all the information of the original signal The signal x(t) may be reconstructed from these samples by the use of a low-pass filter With this theorem, we can know how much high the sampling rate should be [5]

For audio signal, there are two common sampling rates: 8000 Hz and 44100 Hz Note that they are corresponding to the most sensitive range of hearing (4000 Hz) and human limit of hearing (20000 Hz) [5]

2.3.4 Quantization

The signal after sampling is only discrete-time signal, not digital one Because the digital computer can only work with discrete values, we must do another step to convert the continuous range of discrete-time signal into discrete range That step is called quantization Contrast to sampling, where we can preserve perfectly any sinusoidal components with frequency lower than half sampling rate, quantization error cannot be removed entirely, because we cannot expect every amplitude value at sample points is falling exactly into our discrete range Quantization error is also called as quantization noise because of its random nature

For example, we use a 16 bit integer to represent the amplitude of a signal That means our discrete range will contain 65536 equally spaced values The maximum quantization error at one sample is not greater than ± 1 2⁄ 𝐿𝑆𝐵 LSB is least significant bit, which means the amplitude value represented by one least significant bit It also is the amplitude distance between adjacent quantization levels The quantization error at a sample can be considered a random variable uniformly distributed over ± 1 2⁄ 𝐿𝑆𝐵 Therefore, the total quantization error can be thought as an additive noise and quantization is the process that adds this noise into the discrete-time signal

This model is very powerful, because the quantization noise is just added into whatever noise has already existed in the original signal If the existing noise is high compare to quantization noise, then the presence of quantization noise will be insignificant Thus, we only need to increase the quantization resolution until the quantization noise is smaller than existing noise

2.4 Fourier Transform and Frequency domain representation

Normally, the audio signal is represented as a function of time However, this form of representation sometime is hard to work with For example, we want to increase the amplitude of the 1 𝑘𝐻𝑧 sinusoidal component of one particular audio signal 𝑥(𝑡) It is very complicated to calculate directly with 𝑥(𝑡) A wise method would be decomposing 𝑥(𝑡) into sinusoidal component, modifying on each component, and then synthesizing them back into 𝑥(𝑡) That is the basis of Fourier Transform and frequency domain representation

With a signal represented as a function of time:

𝑥(𝑡) Decompose it into sinusoidal components:

cos(2𝜋𝑓𝑡) =𝑒𝑖2𝜋𝑓𝑡+ 𝑒−𝑖2𝜋𝑓𝑡

2sin(2𝜋𝑓𝑡) =𝑒𝑖2𝜋𝑓𝑡− 𝑒−𝑖2𝜋𝑓𝑡

2𝑖𝑥(𝑡) = ∫ (𝑎(𝑓)𝑒𝑖2𝜋𝑓𝑡+ 𝑒−𝑖2𝜋𝑓𝑡

𝑎(𝑓)

𝑏(𝑓)2𝑖 ) 𝑒−𝑖2𝜋𝑓𝑡) 𝑑𝑓

𝑥(𝑡) : − ∞ < 𝑡 < ∞ 𝑋(𝑓): − ∞ < 𝑓 < ∞ 𝑋(𝑓) = 𝑐(𝑓)

𝑥(𝑡) = ∫ 𝑐(𝑓)

∞

−∞

𝑒𝑖2𝜋𝑓𝑡𝑑𝑓 The question is how to calculate 𝑐(𝑓) The mathematicians had determined that:

In signal processing term, we call 𝑋(𝑓) is the frequency spectrum of audio signal 𝑥(𝑡) The value of 𝑋(𝑓) characterize both the amplitude and phase of the sinusoidal component with frequency 𝑓

When we represent the complex value 𝑋(𝑓) in polar form, its absolute value is the amplitude

of sinusoidal component; its argument is the phase of sinusoidal component Therefore, we also call |𝑋(𝑓)| the magnitude spectrum and ∠𝑋(𝑓) the phase spectrum The frequency spectrum can be considered as the combination of magnitude and phase spectrum The benefit of those two spectrums is that they are real signal, not complex, so it is more convenient to work with them

The above is only the Fourier transform for aperiodic continuous signals, for other kind of signals, the Fourier Transform is different: [6]

Table 1: Different kinds of signal and their Fourier Transforms

Periodic continuous with

fundamental frequency 𝑓 and

fundamental period 𝑇

𝑥(𝑡): 0 ≤ 𝑡 ≤ 𝑇 𝑋[𝑘]: … , −2, −1, 0, 1, 2, …

𝑋[𝑘] =1

𝑇∫ 𝑥(𝑡)𝑒−𝑖2𝜋𝑘𝑓𝑡

𝑇 0

𝑑𝑡

𝑥(𝑡) = ∑ 𝑋[𝑘]𝑒𝑖2𝜋𝑘𝑓𝑡

∞ 𝑘=−∞

Aperiodic discrete

𝑥[𝑛]: … , −2, −1, 0, 1, 2, … 𝑋(𝜔): −𝜋 ≤ 𝑓 ≤ 𝜋 𝑋(𝜔) = ∑ 𝑥[𝑛]𝑒 −𝑖𝜔𝑛

∞ 𝑛=−∞

𝑋[𝑘] = 1

𝑁∑ 𝑥[𝑛]𝑒−𝑖𝜔𝑛

𝑁−1 𝑛=0

𝑥[𝑛] = ∑ 𝑋[𝑘]𝑒 𝑖𝜔𝑛

𝑁−1 𝑛=0