VIEWS: 5 PAGES: 44 POSTED ON: 6/2/2012
Signals and Linear System Fourier Transforms Sampling Theorem Fourier Transform Is the extension of Fourier series to non-periodic signal Definition of Fourier transform Fourier transform X ( f ) F[ x(t )] x(t )e j 2 ft dt Inverse Fourier transform x(t ) F 1[ X ( f )] X ( f )e j 2 ft df From Fourier series (T0 ) 1 T xn x(t )e j 2 nf t dt x(t ) xn e j 2 nf t 0 0 0 T0 n Properties of FT For a real signal x(t) X(f) is Hermitian Symmetry X ( f ) X * ( f ) Magnitude spectrum is even about the origin (f=0) Phase spectrum is odd about the origin f, called frequency (units of Hz), is just a parameter of FT that specifies what frequency we are interested in looking for in the x(t) The FT looks for frequency f in the x(t) over - < t < F(f) can be complex even though x(t) is real If x(t) is real, then Hermitian symmetry Properties of FT Linearity F[ x1 (t ) x2 (t )] F[ x1 (t )] F[ x2 (t )] Duality If X ( f ) F[ x(t )] , Then F[ X (t )] x( f ) Time Shift A shift in the time domain results in a phase shift in the frequency domain F ( x(t t )) e X ( f ) ( e X ( f ) X ( f ) ) 0 j 2 ft0 j 2 ft0 Properties of FT Scaling An expansion in the time domain results in a contraction in the frequency domain, and vice versa 1 f F [ x(at )] X( ) , a 0 a a Properties of FT Modulation Multiplication by an exponential in the time domain corresponds to a frequency shift in the frequency domain F[e j 2 f0t x(t )] X ( f f 0 ) 1 F[ x(t ) cos(2 f 0t )] [ X ( f f 0 ) X ( f f 0 )] 2 x(t) cos X(f) -f0 f0 -f0 f0 Properties of FT Differentiation Differentiation in the time domain corresponds to multiplication by j2f in the frequency domain d F[ x(t )] j 2 f X ( f ) dt dn F [ n x(t )] ( j 2 f ) n X ( f ) dt Properties of FT Convolution Convolution in the time domain is equivalent to multiplication in the frequency domain, and vice versa F[ x(t ) y(t )] X ( f )Y ( f ) F[ x(t ) y(t )] X ( f ) Y ( f ) Properties of FT Parseval’s relation x(t ) y* (t )dt X ( f )Y * ( f )df Energy can be evaluated in the frequency domain instead of the time domain Rayleigh’s relation x(t ) x (t )dt x(t ) dt * 2 2 X ( f ) df More on FT pairs See Table 1.1 at page 20 Delta function Flat Time / Frequency shift Sin / cos input Periodic signal impulses in the frequency domain sgn / unit step input Rectangular sinc Lambda sinc2 Differentiation Pulse train with period T0 Periodic signal impulses in the frequency domain FT of periodic signals For a periodic signal with period T0 x(t) can be expressed with FS coefficient x(t ) xe n n j 2 nt / T0 Take FT X ( f ) F [ x(t )] F[ xn e j 2 nt / T0 ] n n n xn F [e j 2 nt / T0 ] n xn ( f T0 ) FT of periodic signal consists of impulses at harmonics of the original signal FS with Truncated signal FS coefficient can be expressed using FT Define truncated signal T0 T0 , t x(t ) xT0 (t ) 2 2 0 , otherwise FT of truncated signal: X T0 ( f ) F[ xT0 (t )] Expression of FS coefficient 1 n xn X T0 ( ) T0 T0 Spectrum of the signal Fourier transform of the signal is called the Spectrum of the signal Generally complex Magnitude spectrum Phase spectrum Illustrative problem 1.5 Time shifted signal Same magnitude, Different phase Try it by yourself with Matlab ! Sampling Theorem Basis for the relation between continuous- time signal and discrete-time signals A bandlimited signal can be completely described in terms of its sample values taken at intervals Ts as long as Ts 1/(2W) x(t) X(f) 1 -W W f Ts Impulse sampling Sampled waveform x (t ) x(t ) (t nTs ) x(nT ) (t nT ) s s n n Take FT X ( f ) F [ x (t )] F [ x(t ) (t nTs )] X ( f ) F [ (t nTs )] n n 1 n 1 n X ( f ) Ts n (f ) Ts Ts n X(f Ts ) x(t) X(f) 1/Ts f -W W 1/Ts Ts 1/(2Ts) Reconstruction of signal Low pass filter With Bandwidth 1/(2Ts) and Gain of Ts X(f) Low pass filter f -W W 1/Ts 1/(2Ts) Reconstruction from sampled signal If we have sampled values {… x(-2Ts), x(-Ts), 0, x(Ts), x(2Ts) …} With Nyquist interval (or Nyquist rate) Ts = 1/(2W) Then we can reconstruct x(t) using x(t ) x(nT )sinc(2W (t nT )) n s s Example: Figure 1.17 at page 24 Aliasing or Spectral folding If sampling rate is Ts > 1/(2W) Spectrum is overlapped We can not reconstruct original signal with under-sampled values Anti-aliasing methods are needed X(f) X(f) 1/Ts f -W W f -W W 1/Ts Discrete Fourier Transform DFT of discrete time sequence x[n] X d ( f ) x[n]e j 2 fnT s n Relation between FT and DFT X ( f ) Ts X d ( f ), for f W FFT(Fast Fourier Transform) Efficient numerical method to compute DFT See fft.m and fftseq.m, for more information Example: Try Illustrative problem 1.6 by yourself DFT in Matlab fft.m and ifft.m with finite samples N Definition of DFT: X (n) x[k ]e j 2 nk / N k 1 1 N Definition of IDFT: x[k ] X (n)e j 2 nk / N N n 1 Time and frequency is not appeared explicitly Just definition implemented on a computer to compute N values for the DFT and IDFT N is chosen to be N=2m Zero padding is used if N is not power of 2 FFT in matlab A sequence of length N=2m of samples of x(t) taken at Ts Ts satisfies Nyquist condition Ts is called time resolution FFT gives a sequence of length N of sampled Xd(f) in the frequency interval [0, fs=1/Ts] The samples are apart f fs / N f is called frequency resolution Frequency resolution is improved by increasing N Some remarks on short signal FT works on signals of infinite duration But, we only measure the signal for a short time FFT works as if the data is periodic all the time Some remarks on short signal Sometimes this is correct Some remarks on short signal Sometimes wrong Frequency leakage If the period exactly fits the measurement time, the frequency spectrum is correct If not, frequency spectrum is incorrect It is broadened Frequency domain analysis of LTI system The output of LTI system y(t ) x(t ) h(t ) Take FT (using convolution theorem) Y ( f ) X ( f )H ( f ) Where the Transfer Function of the system H ( f ) F[h(t )] h(t )e j 2 ft dt The relation between input-output spectra Y( f ) X ( f ) H( f ) Y ( f ) X ( f ) H ( f ) Homeworks Illustrative problem 1.7 Problems 1.10, 1.12, 1.14, 1.15 More on Sampling Most real signals are analog The analog signal has to be converted to digital Information lost during this procedure (Quantization Error) Inaccuracies in measurement Uncertainty in timing Limits on the duration of the measurement More on Sampling Continuous analog signal has to be held before it can be sampled More on Sampling The sampling take place at equal interval of time after the hold Need fast ADC Need fast hold circuit Signal is not changing during the time the circuit is acquiring the signal value Unless, ADC has all the time that the signal is held to make its conversion We don’t know what we don’t measure More on Sampling In the process of measuring signal, some information is lost Aliasing We only sample the signal at intervals We don’t know what happens between the samples Aliasing We must sample fast enough to see the most rapid changes in the signal This is Sampling theorem If we do not sample fast enough Some higher frequencies can be incorrectly interpreted as lower ones Aliasing Called “aliasing” because one frequency looks like another Aliasing Nyquist frequency We must sample faster than twice the frequency of the highest frequency component Antialiasing We simply filter out all the high frequency components before sampling Antialias filters must be analog It is too lte once you have done the sampling More on sampling Theorem The sampling theorem does not say the samples will look like the signal More on Sampling Theorem Sampling theorem says there is enough information to reconstruct the signal Correct reconstruction is not just draw straight lines between samples More on Sampling Theorem The impulse response of the reconstruction filter has sinc (sinx/x) shape The input to the filter is the series of discrete impulses which are samples Every time an impulse hits the filter, we get ‘ringing’ Superposition of all these rings reconstruct the proper signal Frequency resolution We only sample the signal for a certain time We must sample for at least one complete cycle of the lowest frequency we want to resolve Quantization When the signal is converted to digital form Precision is limited by the number of bits available Uncertainty in the clock Uncertainty in the clock timing leads to errors in the sampled signal Digitization errors The errors introduced by digitization are both nonlinear and signal dependent Nonlinear We can not calculate their effect using normal maths. Signal dependent The errors are coherent and so cannot be reduced by simple means Digitization errors The effect of quantiztion error is often similar to an injection of random noise