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AI & Speech Processing: DSP-1

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본 글은 광운대학교 전자공학과 박호종 교수님의 강의를 듣고 작성되었음을 밝힙니다.

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DSP Review

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Time-to-Frequency transform

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System Equation

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CTFT and DTFT

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Continuous-Time Fourier Transform (CTFT)

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\"image\"

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basis signal : 주파수를 정의하는 가장 기본적인 식 (f0는 주파수)

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correlation : 2개의 시그널을 곱하고 전체 적분을 취하면 그것이 곧 두 신호의 비슷한 정도를 나타낸다.

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x(t)에 f0가 얼마나 들어있는지를 correlation으로 측정한다.
\n(Conjucate를 취해줘서 -j2πft로 바뀜)

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역변환의 관계를 이용한 식

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Conversion from x(t) to x[n]

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\"image\"

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    \n
  1. \n

    Analog Signal을 주기 T마다 Sampling을 한다.

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    2. \n
  2. \n

    축을 t에서 n으로 변환시킨다.

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    3. \n
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주의할 점은 n축일때에 1이 1초일때가 아니라는 점.
\n=> 우리는 샘플 주기를 모른다면, 몇초인지를 알 수 있다.

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Discrete-Time Fourier Transform (DTFT)

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\"image\"

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시간 축에서 T마다 샘플링을 하게 되면, 기존 스펙트럼이 1 / T 를 주기로 반복되고 크기는 1 / T로 줄어든다.

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1 / T는 주파수를 의미하기도 한다.
\n=> 1초에 샘플을 몇개 얻어오느냐

\n\n

주기 T가 1로 1 / T배 줄어들게되면, 스펙트럼은 반대로 T배만큼 늘어나게 된다.

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3번째 그림의 f는 주파수가 아닌 cucle per sample이라는 새로운 f임 주의 !!

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중간과정을 생략한채 보게 되면, 기존 스펙트럼이 T배 벌어지고, 1마다 반복되게 된다.

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(a가 Ta가 되고, 1마다 반복된다)

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3번째 그림의 1을 가르키면서 몇 Hz냐고 묻는다면, Sampling 주파수에 해당한다 !!

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Conclusion

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결국 CTFT와 DTFT는 같은 것이지만, Continuous하냐, Discrete하냐에 따라서 Sigma를 해주느냐, Summation을 해주느냐만이 다른 것이다.

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\"image\"

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Response"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"text","value":"z-transform"}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"h2","properties":{},"children":[{"type":"text","value":"CTFT and DTFT"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"h3","properties":{},"children":[{"type":"text","value":"Continuous-Time Fourier Transform (CTFT)"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"element","tagName":"img","properties":{"src":"https://user-images.githubusercontent.com/42150335/78560266-d08f7200-7850-11ea-930a-c4dc16410ccf.png","alt":"image"},"children":[]}]},{"type":"text","value":"\n"},{"type":"element","tagName":"ul","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"ω = 2πf"}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"X(f) : Spectrum"}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"x(t) : Signal"}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"X(f)식 해석"}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"element","tagName":"strong","properties":{},"children":[{"type":"text","value":"basis signal"}]},{"type":"text","value":" : 주파수를 정의하는 가장 기본적인 식 (f0는 주파수)"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"element","tagName":"strong","properties":{},"children":[{"type":"text","value":"correlation"}]},{"type":"text","value":" : 2개의 시그널을 곱하고 전체 적분을 취하면 그것이 곧 두 신호의 비슷한 정도를 나타낸다."}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"x(t)에 f0가 얼마나 들어있는지를 correlation으로 측정한다."},{"type":"element","tagName":"br","properties":{},"children":[]},{"type":"text","value":"\n(Conjucate를 취해줘서 -j2πft로 바뀜)"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"ul","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"text","value":"x(t)식 해석"}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"역변환의 관계를 이용한 식"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"h3","properties":{},"children":[{"type":"text","value":"Conversion from x(t) to x[n]"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"element","tagName":"img","properties":{"src":"https://user-images.githubusercontent.com/42150335/78560292-ddac6100-7850-11ea-8c23-e7da693d1bfd.png","alt":"image"},"children":[]}]},{"type":"text","value":"\n"},{"type":"element","tagName":"ol","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"Analog Signal을 주기 T마다 Sampling을 한다."}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"축을 t에서 n으로 변환시킨다."}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"주의할 점은 n축일때에 1이 1초일때가 아니라는 점."},{"type":"element","tagName":"br","properties":{},"children":[]},{"type":"text","value":"\n=> 우리는 샘플 주기를 모른다면, 몇초인지를 알 수 있다."}]},{"type":"text","value":"\n"},{"type":"element","tagName":"h3","properties":{},"children":[{"type":"text","value":"Discrete-Time Fourier Transform (DTFT)"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"element","tagName":"img","properties":{"src":"https://user-images.githubusercontent.com/42150335/78560748-86f35700-7851-11ea-865e-7946c84988a6.png","alt":"image"},"children":[]}]},{"type":"text","value":"\n"},{"type":"element","tagName":"ul","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"element","tagName":"strong","properties":{},"children":[{"type":"text","value":"중요 포인트 1"}]}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"시간 축에서 T마다 샘플링을 하게 되면, 기존 스펙트럼이 1 / T 를 주기로 반복되고 크기는 1 / T로 줄어든다."}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"1 / T는 주파수를 의미하기도 한다."},{"type":"element","tagName":"br","properties":{},"children":[]},{"type":"text","value":"\n=> 1초에 샘플을 몇개 얻어오느냐"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"ul","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"element","tagName":"strong","properties":{},"children":[{"type":"text","value":"중요 포인트 2"}]}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"주기 T가 1로 1 / T배 줄어들게되면, 스펙트럼은 반대로 T배만큼 늘어나게 된다."}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"3번째 그림의 f는 주파수가 아닌 cucle per sample이라는 새로운 f임 주의 !!"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"ul","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"element","tagName":"strong","properties":{},"children":[{"type":"text","value":"중요 포인트 3"}]}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"중간과정을 생략한채 보게 되면, 기존 스펙트럼이 T배 벌어지고, 1마다 반복되게 된다."}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"(a가 Ta가 되고, 1마다 반복된다)"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"ul","properties":{},"children":[{"type":"text","value":"\n"},{"type":"element","tagName":"li","properties":{},"children":[{"type":"element","tagName":"strong","properties":{},"children":[{"type":"text","value":"문제 1"}]}]},{"type":"text","value":"\n"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"3번째 그림의 1을 가르키면서 몇 Hz냐고 묻는다면, Sampling 주파수에 해당한다 !!"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"h3","properties":{},"children":[{"type":"text","value":"Conclusion"}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"text","value":"결국 CTFT와 DTFT는 같은 것이지만, Continuous하냐, Discrete하냐에 따라서 Sigma를 해주느냐, Summation을 해주느냐만이 다른 것이다."}]},{"type":"text","value":"\n"},{"type":"element","tagName":"p","properties":{},"children":[{"type":"element","tagName":"img","properties":{"src":"https://user-images.githubusercontent.com/42150335/78562135-d9ce0e00-7853-11ea-8fa4-64d5969a5a3d.png","alt":"image"},"children":[]}]}],"data":{"quirksMode":false}},"excerpt":"AI & Speech Processing: DSP-1 본 글은 광운대학교 전자공학과 박호종 교수님의 강의를 듣고 작성되었음을 밝힙니다. DSP Review Time-to-Frequency transform Continuous-Time Fourier…","fields":{"readingTime":{"text":"3 min read"}},"frontmatter":{"title":"Sooftware Speech - AI & Speech Processing: DSP-1","userDate":"8 April 2020","date":"2020-04-08T10:00:00.000Z","tags":["speech","dsp"],"excerpt":null,"image":{"childImageSharp":{"gatsbyImageData":{"layout":"fullWidth","backgroundColor":"#f8f8f8","images":{"fallback":{"src":"/static/b2a62b0f00909ef6d1e35130041e84ea/ef400/dsp1.png","srcSet":"/static/b2a62b0f00909ef6d1e35130041e84ea/ef400/dsp1.png 727w","sizes":"100vw"},"sources":[{"srcSet":"/static/b2a62b0f00909ef6d1e35130041e84ea/e8edc/dsp1.webp 727w","type":"image/webp","sizes":"100vw"}]},"width":1,"height":0.3246217331499312}}},"author":[{"id":"Soohwan Kim","bio":"Co-founder/A.I. engineer at TUNiB.","avatar":{"children":[{"gatsbyImageData":{"layout":"fullWidth","backgroundColor":"#282838","images":{"fallback":{"src":"/static/a9e6b445142b247ee4cfa66155398bb2/0d6f4/soohwan.png","srcSet":"/static/a9e6b445142b247ee4cfa66155398bb2/248f9/soohwan.png 40w,\n/static/a9e6b445142b247ee4cfa66155398bb2/fd435/soohwan.png 80w,\n/static/a9e6b445142b247ee4cfa66155398bb2/0d6f4/soohwan.png 120w","sizes":"100vw"},"sources":[{"srcSet":"/static/a9e6b445142b247ee4cfa66155398bb2/e7f45/soohwan.webp 40w,\n/static/a9e6b445142b247ee4cfa66155398bb2/589ec/soohwan.webp 80w,\n/static/a9e6b445142b247ee4cfa66155398bb2/71a38/soohwan.webp 120w","type":"image/webp","sizes":"100vw"}]},"width":1,"height":0.6833333333333333}}]}}]}},"relatedPosts":{"totalCount":20,"edges":[{"node":{"id":"fa9e8cbb-841a-516f-9df6-4be0336b56b0","excerpt":"한국어 Tacotron2 이번 포스팅에서는 Tacotron2 아키텍처로 한국어 TTS 시스템을 만드는 방법에 대해 다루겠습니다. 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DFT (Discrete Fourier Transform) Digital 처리를 위하여 time와 frequency…","frontmatter":{"title":"Sooftware Speech - AI & Speech Processing: DSP-2","tags":["speech","dsp"],"date":"2020-04-09T10:00:00.000Z","draft":false,"excerpt":null,"image":{"childImageSharp":{"gatsbyImageData":{"layout":"fullWidth","placeholder":{"fallback":"data:image/png;base64,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"},"images":{"fallback":{"src":"/static/469a97adb92a59804305e435dac71520/cb4cb/dsp2.png","srcSet":"/static/469a97adb92a59804305e435dac71520/e6cc4/dsp2.png 750w,\n/static/469a97adb92a59804305e435dac71520/c8e41/dsp2.png 1080w,\n/static/469a97adb92a59804305e435dac71520/3f384/dsp2.png 1366w,\n/static/469a97adb92a59804305e435dac71520/cb4cb/dsp2.png 1373w","sizes":"100vw"},"sources":[{"srcSet":"/static/469a97adb92a59804305e435dac71520/79bef/dsp2.webp 750w,\n/static/469a97adb92a59804305e435dac71520/c28c7/dsp2.webp 1080w,\n/static/469a97adb92a59804305e435dac71520/a040e/dsp2.webp 1366w,\n/static/469a97adb92a59804305e435dac71520/668c2/dsp2.webp 1373w","type":"image/webp","sizes":"100vw"}]},"width":1,"height":0.5105608157319738}}},"author":[{"id":"Soohwan Kim","bio":"Co-founder/A.I. engineer at TUNiB.","avatar":{"childImageSharp":{"gatsbyImageData":{"layout":"fullWidth","placeholder":{"fallback":"data:image/png;base64,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"},"images":{"fallback":{"src":"/static/a9e6b445142b247ee4cfa66155398bb2/7cf1f/soohwan.png","srcSet":"/static/a9e6b445142b247ee4cfa66155398bb2/34f77/soohwan.png 750w,\n/static/a9e6b445142b247ee4cfa66155398bb2/a94f6/soohwan.png 1080w,\n/static/a9e6b445142b247ee4cfa66155398bb2/7cf1f/soohwan.png 1148w","sizes":"100vw"},"sources":[{"srcSet":"/static/a9e6b445142b247ee4cfa66155398bb2/38420/soohwan.webp 750w,\n/static/a9e6b445142b247ee4cfa66155398bb2/7470d/soohwan.webp 1080w,\n/static/a9e6b445142b247ee4cfa66155398bb2/b5ef6/soohwan.webp 1148w","type":"image/webp","sizes":"100vw"}]},"width":1,"height":0.6829268292682927}}}}]},"fields":{"readingTime":{"text":"10 min read"},"layout":"","slug":"/dsp2/"}},"primaryTag":"speech"}}, "staticQueryHashes": ["3170763342","3229353822"]}