Fundamentals of Measurement Technology (4) Prof Wang Boxiong
Fundamentals of Measurement Technology (4) Prof. Wang Boxiong
2.3 Digital signal processing aDigital signal processing, a field which has its roots in 17th and 18th century mathematics has become an important modern tool in a multitude of diverse fields of science and technology ODigital signal processing is concerned with the representation of signals by sequences of numbers or symbols and the processing of these sequences
❑Digital signal processing, a field which has its roots in 17th and 18th century mathematics, has become an important modern tool in a multitude of diverse fields of science and technology. ❑Digital signal processing is concerned with the representation of signals by sequences of numbers or symbols and the processing of these sequences. 2.3 Digital signal processing
2. 3 Digital signalprocessing UThe availability of high speed digital computers has fostered the development of increasingly complex and sophisticated signal processing algorithms, and recent advances in integrated circuit technology promise economical implementations of very complex digital processing systems OThe evolution of a new point of view toward digital signal processing was further accelerated by the disclosure in 1965 of an efficient algorithm for computation of Fourier transforms the fast fourier transform or fft
❑The availability of high speed digital computers has fostered the development of increasingly complex and sophisticated signal processing algorithms, and recent advances in integrated circuit technology promise economical implementations of very complex digital processing systems. ❑The evolution of a new point of view toward digital signal processing was further accelerated by the disclosure in 1965 of an efficient algorithm for computation of Fourier transforms: the fast Fourier transform or FFT. 2.3 Digital signal processing
2. 3 Digital signalprocessing UThe fast Fourier transform algorithm reduced the computation time of Fourier transform by orders of magnitude UThe importance of digital signal processing appears to be increasing with no visible sign of saturation OThe impact of digital signal processing techniques will undoubtedly promote revolutionary advances in some fields of application
❑The fast Fourier transform algorithm reduced the computation time of Fourier transform by orders of magnitude. ❑The importance of digital signal processing appears to be increasing with no visible sign of saturation. ❑The impact of digital signal processing techniques will undoubtedly promote revolutionary advances in some fields of application. 2.3 Digital signal processing
2.3.1 Discrete Fourier Transform ( DFT) For a nonperiodic continuous time signal X(), its Fourier transform must be a continuous spectrum X( FT:X(=x()e 2nf (2199) IFT: x(= X(k/mdf (2200) The continuous time signals and the continuous spectra must be discretized first and then truncated to get a finite length of sequence before being processed by a computer. This forms just the basis for the discrete Fourier transform(DFT)
For a nonperiodic continuous time signal x(t), its Fourier transform must be a continuous spectrum X(f), The continuous time signals and the continuous spectra must be discretized first and then truncated to get a finite length of sequence before being processed by a computer. This forms just the basis for the discrete Fourier transform (DFT). 2.3.1 Discrete Fourier Transform (DFT) ( ) ( ) − − FT X f = x t e dt j2ft : (2.199) IFT x(t) X(f )e df j2ft : − = (2.200)
2.3. 1 Discrete Fourier Transform DFT There are four cases for the fourier transform of an infinite-length continuous signal(Fig 263) mmmcaeT 2-20T/2 2072J。了 x () 灬A Fig 2.63 Types of Fourier transform
There are four cases for the Fourier transform of an infinite-length continuous signal (Fig. 2.63). 2.3.1 Discrete Fourier Transform (DFT) Fig. 2.63 Types of Fourier transform
2.3. 1 Discrete Fourier Transform DFT) uFig 2.63(a): a nonperiodic continuous signal x(t)and its Fourier transform spectrum X(. The spectrum is continuous OFig. 2.63 (b): a periodic continuous signal and the frequency spectrum is or discrete F:X()=「x(012dt (2201) JFT:x()=∑X()e2m (2202) where f=ky(k=0,±1±2…) Af: fundamental frequencys
❑Fig. 2.63 (a): a nonperiodic continuous signal x(t) and its Fourier transform spectrum X(f). The spectrum is continuous. ❑Fig. 2.63 (b): a periodic continuous signal, and the frequency spectrum is or discrete. where Δf: fundamental frequency, 2.3. 1 Discrete Fourier Transform (DFT) ( ) ( ) − − = 2 2 1 2 : T T j f kt k x t e dt T FT X f (2.201) ( ) ( ) =− = k j f t k k IFT x t X f e 2 : (2.202) f = kf (k = 0,1,2, ) k T f 1 =
2. 3. 1 Discrete Fourier transform DFT uFig 2.63(c): the Fourier transform of a nonperiodic discrete signal The Fourier transform of an infinite-length discrete time sequence is a periodic continuous spectrum FT:X()=∑ x(t, e2m (2203) IFT:x(t, 乙X(/2d 2.204) Whee〃sn△(n=0,±1+2…) At is the sampling period; fs is the sampling frequency of the time sequence
❑Fig. 2.63 (c): the Fourier transform of a nonperiodic discrete signal. The Fourier transform of an infinite-length discrete time sequence is a periodic continuous spectrum. where Δt is the sampling period; fs is the sampling frequency of the time sequence. 2.3.1 Discrete Fourier Transform (DFT) ( ) ( ) =− − = n j f t n n FT X f x t e 2 : (2.203) ( ) ( ) − = 2 2 1 2 : s s n f f j ft s n X f e df f IFT x t (2.204) t = nt(n = 0,1,2, ) n s f t 1 =
2.3. 1 Discrete Fourier Transform DFT uFig 2.63(d): the Fourier transform of a periodic discrete time sequence. Its spectrum is also periodic and discrete The sampling period is At, then T=N△t
❑Fig. 2.63 (d): the Fourier transform of a periodic discrete time sequence. Its spectrum is also periodic and discrete. The sampling period is Δt, then 2.3.1 Discrete Fourier Transform (DFT) T = Nt
2. 3. 1 Discrete Fourier Transform DFT) Conclusion: For a periodic x(t the spectrum X( is bound to be discrete and vice versa ☆lfx() is nonperiodic,then×(is continuous, and vice versa the case shown in Fig. 2. 64 (d)where both the time and frequency signals are discrete and periodic provides us with the possibility of using a computer to implement spectrum analysis
Conclusion: ❖For a periodic x(t) the spectrum X(f) is bound to be discrete, and vice versa. ❖If x(t) is nonperiodic, then X(f) is continuous, and vice versa. ❖The case shown in Fig. 2.64 (d) where both the time and frequency signals are discrete and periodic provides us with the possibility of using a computer to implement spectrum analysis. 2.3.1 Discrete Fourier Transform (DFT)
