The Fourier Transform (and more )

Transcription

1 The Fourier Transform (and more ) imrod Peleg ov. 5 Outline Introduce Fourier series and transforms Introduce Discrete Time Fourier Transforms, (DTFT) Introduce Discrete Fourier Transforms (DFT) Consider operational complexity of DFT Deduce a radix- FFT algorithm Consider some implementation issues of FFTs with DSPs

2 The Frequency Domain The frequency domain does not carry any information that is not in the time domain. The power in the frequency domain is that it is simply another way of looing at signal information. Any operation or inspection done in one domain is equally applicable to the other domain, except that usually one domain maes a particular operation or inspection much easier than in the other domain. frequency domain information is extremely important and useful in signal processing. 3 3 Basic Representations for FT. An Exponential Form. A Combined Trigonometric Form A Simple Trigonometric Form

3 The Fourier Series: Exponential Form Periodic signal expressed as infinite sum of sinusoids. Complex umbers! x (t) = c p = T p Tp c = x p e jω t (t)e, jω t where dt C s are frequency domain amplitude and phase representation For the given value x p (t) (a square value), the sum of the first four terms of trigonometric Fourier series are: x p (t). + sin(t) + C sin(3t) + C 3 sin(5t) 5 The Combined Trigonometric Form Periodic signal: x p (t) = x p (t+t) for all t and cycle time (period) is: T π = = f ω f is the fundamental frequency in Hz w is the fundamental frequency in radians: ω = π f x p(t) can be expressed as an infinite sum of orthogonal functions. When these functions are the cosine and sine, the sum is called the Fourier Series. The frequency of each of the sinusoidal functions in the Fourier series is an integer multiple of the fundamental frequency. Basic frequency + Harmonies 6 3

4 Fourier Series Coefficients j t e ω Each individual term of the series, C,is the frequency domain representation and is generally complex (frequency and phase), but the sum is real. The second common form is the combined trigonometric form: xpt () = C + Csin( ωt + θ) = C θ = tan Im( ) Re( C) Again: C are Complex umbers! 7 The Trigonometric Form pt () = + ( ω )+ ( ω ) x A ACos t BSin t = A = x dt= DC = Average value of x p( t) p( t) Tp T p A= xp() t Cos( ω t) dt Tp = x in( ω ) B p() t S t dt Tp All three forms are identical and are related using Euler s identity: ± jθ e = Cosθ ± jsinθ Thus, the coefficients of the different forms are related by: C = A jb ; C= A B C = θ tan = tan Im( ) A Reminder: e + e Cos θ = jθ -jθ e -e Sinθ = j jθ -jθ Re( C ) 8

5 The Fourier Transform /3 The Fourier series is only valid for periodic signals. For non-periodic signals, the Fourier transform is used. Most natural signals are not periodic (speech). We treat it as a periodic waveform with an infinite period. If we assume that T P tends towards infinity, then we can produce equations ( model ) for non-periodic signals. If Tp tends towards infinity, then w tends towards. Because of this, we can replace w with dw, and it leads us to: dω fp lim = p t Tp π T = = ω π 9 The Fourier Transform /3 C dω z ( ω ) = ( ) π jωt x t e dt Increase T P = Period Increases : o Repetition: ω dω = Tp π π Discrete frequency variable becomes continuous: ω ω Discrete coefficients C become continuous: C C ( ω ) C ( ω ) π ω d = jωt x() t e dt 5

6 The Fourier Transform 3/3 C( ω) dω We define: π X ( ω) = F{ xt ( )} z jωt jωt X ( ω ) = x( t) e dt x() t = x( ω ) e dω π z Signal Representation by Delta Function Instead of a continuous signal we have a collection of samples : This is equivalent to sampling the signal with one Delta Function each time, moving it along X-axis, and summing all the results: x st () () t ( t nts) = x δ ote that the Delta is only If its index is zero! 6

7 Discrete Time Fourier Transform /3 Consider a sampled version, x s (t), of a continuous signal, x(t) : xst () = x() tδ ( t nts) Ts is the sample period. We wish to tae the Fourier transform of this sampled signal. Using the definition of Fourier transform of x s(t) and some mathematical properties of it we get: Replace continuous time t with (nt s ) Continuous x(t) becomes discrete x(n) Sum rather than integrate all discrete samples xs( ω ) = x( nts) e n= jωnts 3 Discrete Time Fourier Transform /3 Fourier Transform ω t j x( ω ) = x( t) e dt x( Ω) = x(n)e n= j Ω n Discrete Time Fourier Transform Inverse Fourier Transform = x(t) π x( ω)e ωt j dω j x(n) = Ω π x( )e π π (nω ) dω Inverse Discrete Time Fourier Transform Limits of integration need not go beyond ±π because the spectrum repeats itself outside ±π (every π): X ( Ω) = X ( Ω + π ) Keep integration because X ( Ω) is continuous: Ω = ωts means that Ω is periodic every Ts 7

8 Discrete Time Fourier Transform 3/3 ow we have a transform from the time domain to the frequency domain that is discrete, but... DTFT is not applicable to DSP because it requires an infinite number of samples and the frequency domain representation is a continuous function impossible to represent exactly in digital hardware. 5 st result: yquist Sampling Rate / The Spectrum of a sampled signal is periodic, with *Pi Period: X( Ω ) = X( Ω+ π ) Easy to see: ( Ω+ π ) Ω X( Ω+ π ) = x( n) e = x( n) e e n= jnω = xne ( ) = X( Ω) n= jn jn j π n n= jπ n e πn j πn = cos( ) sin( ) = 6 8

9 st result: yquist Sampling Rate / For maximum frequency w H : Ω = H = ω ω π Ω= ωts π = ωhts π π Ts = BUT : Ts = ωh ωs π ωs = = ωh Ts 7 yquist Conclusion 8 9

10 Practical DTFT Tae only time domain samples n= Ω n j x ( Ω) = x( n) e x( Ω) = n= x( n) e j Ω n Sample the frequency domain, i.e. only evaluate x(ω) at discrete points. The equal spacing between points is Ω = π/ π πn/ j x( ) = xne ( ) =,,,..., n= 9 The DFT Since the only variable in π / is, the DTFT is written: j x ( ) = xne ( ) =,,,..., n= πn/ Using the shorthand notation: W j = e π / (Twiddle Factor) The result is called Discrete Fourier Transform (DFT): n X( ) = x( n) W and x( n) = X( ) W n= = n

11 Usage of DFT The DFT pair allows us to move between the time and frequency domains while using the DSP. The time domain sequence x[n] is discrete and has spacing Ts, while the frequency domain sequence X[] is discrete and has spacing /T [Hz]. DFT Relationships Time Domain Frequency Domain X(n) Samples x() Samples T s T s 3T s (-)T s 3 - t n / - - F s F s F s s F F s f

12 Standard DFT: Practical Considerations An example of an 8 point DFT: n n n= X ( ) = x ( ) W Writing this out for each value of n : 7 Xn () = x()w7 + x()w x(7)w7, =,,...,7 7 X ( ) = x ( ) W =,,,...,7 n = n 7 Each term such as x()w 7 requires 8 multiplications Total number of (complex!) multiplications required: 8 * 8 = 6 -point DFT requres = 6 complex multiplications And all of these need to be summed. 3 Symmetry Property Fast Fourier Transform W + / = W Periodicity Property Splitting the DFT in two (odd and even) or Manipulating the twiddle factor W = W X + X () = r= x(r).w r r () = x(r).(w ) + W x(r + ).(W ) r= r= π j( ) r π j( / ) r= W = e = e = W + x(r + ). W / (r+ ) X ( ) = n THE FAST FOURIER TRASFORM r = x(r )W r + W r = x(r + ) W r

13 x ( ) = r= FFT complexity x(r) W r / + W x(r + ) r = W r / (/) multiplications (/) multiplications / Multiplications For an 8-point FFT, + + = 36 multiplications, saving 6-36 = 8 multiplications For point FFT, = 5,5 multiplications, saving,, - 5,5 = 95, multiplications 5 Time Decimation Splitting the original series into two is called decimation in time Let us tae a short series where = 8 Decimate once Called Radix- since we divided by n = {,,, 3,, 5, 6, 7}n = {,,, 6 } and {, 3, 5, 7 } Decimate again n = {, } {, 6 } {, 5 } and { 3, 7 } The result is a savings of (/)log multiplications: point DFT =,8,576 multiplications point FFT = 5 multiplication Decimation simplifies mathematics but there are more twiddle factors to calculate, and a practical FFT incorporates these extra factors into the algorithm 6 3

14 Simple example: -Point FFT Let us consider an example where =: Decimate in time into series: n = {, } and {, 3} r X ( ) = x( r) W + W x( r+ ) W X 3 n ( ) = x( n) W r r = r = = {() x + x() W } + W {() x + x() 3 W } We have two twiddle factors. Can we relate them? ow our FFT becomes: W = e π j π π j j W = e = e = W X ( ) = {() x + x() W } + W {() x + x() 3 W } 7 -Point FFT Flow Diagram The DFT s: for =,,,3 X ( ) = {() x + x() W } + W {() x + x() 3 W } For = only: X ( ) = {() x + x() W } + W {() x + x() 3 W } A flow-diagram of it: x () x () W + W + x () W x (3) + This is for only / of the whole diagram! 8

15 A Complete Diagram X ( ) = {() x + x() W } + W {() x + x() 3 W } X ( ) = { x( ) + x( ) W } + W { x( ) + x( 3) W } X ( ) = {() x + x() W } + W {() x + x() 3 W } 3 X 3) ( = { x( ) + x( ) W } + W { x( ) + x( 3) W } x() x() x() x(3) 3 X () X () X () X (3) ote: W e j = π π j W = e = = W 6 π j 6 W = e = = W 9 The Butterfly A Typical Butterfly x X = x + W x X Twiddle Conversions W = W = -j x W X X = x W x W = - W 3 = j Point FFT Equations Point FFT Butterfly X = (x + x ) + W (x +x 3 ) x X X = (x x ) + W (x x 3 ) X = (x + x ) W (x +x 3 ) X 3 = (x x ) W (x x 3 ) x x W W X X x 3 3 X 3 5

16 Summary Frequency domain information for a signal is important for processing Sinusoids can be represented by phasors Fourier series can be used to represent any periodic signal Fourier transforms are used to transform signals From time to frequency domain From frequency to time domain DFT allows transform operations on sampled signals DFT computations can be sped up by splitting the original series into two or more series FFT offers considerable savings in computation time DSPs can implement FFT efficiently 3 Bit-Reversal If we loo at the inputs to the butterfly FFT, we can see that the inputs are not in the same order as the output. To perform an FFT quicly, we need a method of shuffling these input data addresses around to the correct order. This can be done either by reversing the order of the bits that mae up the address of the data, or by pointer manipulation (bit reversed addition). Many DSPs have special addressing modes that allow them to automatically shuffle the data in the bacground. 3 6

17 Bit-Reversal example x() x() x() x(3) 3 X () X () X () X (3) To obtain the output in ascending order the input values must be loaded in the order: {,,,3} for 5 or it is much more complicated point Bit-Reversal Consider a 3-bit address (8 possible locations). After starting at zero, we add half of the FFT length at each address access with carrying from left to right (!) Start at = =x() + = =x() + = =x() + = =x(6) + = =x() + = =x(5) + = =x(3) + = =x(7) ote that reversing the order of the address bits gives same result! 3 7

18 DCT Type II*: And what about DCT??? π ( n+ ) X ( ) xn ( ) b()x () cos II ()= b() x( n)cos n= II π n+ = = if = b()= if =,...,L- The rest of the math is quite similar.. ote: at least types of DCT!!! * after: A course in Digital Signal Processing, Boaz Porat 35 DCT Type II Most used for compression: JPEG, MPEG etc. Type I Type II Type III Type IV DCT Basis Vectors for =8 36 8

19 DCT Features Real transformation Reversible transformation D Transformation exists and separable Better than the DFT as a de-correlator Fast algorithm exists (log Complexity) 37 Aliasing, Transforms and More yquist's Sampling Theorem is explained here: The effect of the above and aliasing is also shown here: nyquist.cfm And examples to aliasing in images is shown here: Java examples of various signal processing: