Discrete-Time Fourier Analysis PDF

Lecture 3 Discrete-Time Fourier Analysis (chapter 3) Review The output y(n) of a linear system to an arbitrary input x(n)  +∞  +∞ y (n) = L[ x(n)] = L  ∑ x(k )δ (n − k ) = ∑ x(k ) L[δ (n − k )]  k =−∞  n =−...

Lecture 3 Discrete-Time Fourier Analysis (chapter 3) Review The output y(n) of a linear system to an arbitrary input x(n)  +∞  +∞ y (n) = L[ x(n)] = L  ∑ x(k )δ (n − k ) = ∑ x(k ) L[δ (n − k )]  k =−∞  n =−∞ L[δ (n − k )] is called impulse response (denoted by h(n,k)) For a LTI system, unit impulse response becomes h(n, k ) = L[δ (n − k )] = h(n − k ) A LTI system can be represented in terms of its response to the unit sample sequence. The convolution representation is based on the fact that any signal can be represented by a linear combination of scaled and delayed unit samples. Representation Also can represent any arbitrary discrete signal as a linear combination of basis signals introduced in Chap 2. Each basis signal set provides a new signal representation. Each representation has some advantages and disadvantages depending upon the type of system under consideration. Consider DTFT: It is based on the complex exponential { } signal set e j ωn Discrete-Time Fourier Transform +∞ If x[n] is absolutely summable, ∑−∞ | x(n) |< ∞ then its DTFT is given by +∞ DTFT: X ( e jw ) = F [ x ( n )] = ∑ x ( n ) e − jwn n = −∞ 1 +π IDTFT: x(n) = F −1[ X (e jw )] = ∫ X ( e jw ) e jwn dw 2π −π F[.] transforms a discrete signal x(n) into a complex-valued continuous function X of real variable w, called a digital frequency, which is measured in radians. Time domain Frequency domain Discrete Continuous Real valued Complex-valued Summation Integral The range of w: − ∞ − +∞ The integral range of w: − π − +π Discrete-Time Fourier Transform +∞ If x[n] is absolutely summable, ∑−∞ | x(n) |< ∞ then its DTFT is given by +∞ DTFT: X ( e jw ) = F [ x ( n )] = ∑ x ( n ) e − jwn n = −∞ 1 +π IDTFT: x(n) = F −1[ X (e jw )] = ∫ X ( e jw ) e jwn dw 2π −π F[.] transforms a discrete signal x(n) into a complex-valued continuous function X of real variable w, called a digital frequency, which is measured in radians. Ex.31: Determine the Discrete-Time Fourier Transform of x(n) = (0.5) n u (n) This seq. is ABS SUM, thus its DTFT exists Two Important Properties Periodicity: The DTFT is periodic in w with period 2pi X (e jw ) = X (e j[ w+ 2π ] ) ω ∈ [0,2π ], or ω ∈ [−π , π ] Implication: we need only one period for analysis and not the whole domain − ∞ < ω < +∞ Symmetry: For real-valued x(n), X is conjugate symmetric. X (e − jw ) = X * (e jw ) Implication: to plot X, we now need to consider only a half of X: [0,pi] Matlab Implementation If x(n) is of infinite duration, then Matlab can not be used directly to compute X from x(n). But we can use it to evaluate the expression X over [0,pi] frequencies and then plot its magnitude & angle (or real & imaginary parts). Evaluate X (e jw ) in exp.3.1 at 501 equi-spaced points between [0,pi] Strongly recommend to plot freq. in the units of pi Matlab Implementation For a finite duration, if we evaluate X (e jw ) at equi-spaced freq. between [0,pi], the DTFT can be implemented as a matrix-vector multiplication operation. When {x(nl )} and {X (e jw )} are arranged as column vectors x k and X respectively, we have X = Wx In Matlab In Matlab, a dtft function was made for this operation. This way of calculation produces N x (M+1) matrix: memory cost DFT FFT for more efficient computation Example 3.5: periodicity / symmetry ? Example 3.6: periodicity / symmetry ? The properties of the DTFT 1. Linearity: The DTFT is a linear transformation. 2. Time shifting: A shift in time domain corresponds to the phase shifting. 3. Frequency shifting: Multiplication by a complex exponential corresponds to a shift in the frequency domain. 4. Conjugation: Conjugation in the time domain corresponds to the folding and conjugation in the frequency domain. The properties of the DTFT 5. Folding: Folding in the time domain corresponds to the folding in the frequency domain. 6. Symmetries in real sequence: Implication: If the sequence x(n) is real and even, then X is also real and even. 7. Convolution: Freq. domain representation of LTI system The Fourier transform representation is the most useful signal representation for LTI systems. It is due to the following result: Response to a complex exponential ejw0n Response to sinusoidal sequences Response to arbitrary sequences Response to a complex exponential e jw0 n h(n) h(n) * e jw0 n = [ F [h(n)] |w= w0 ]e jw0 n Frequency response: the DTFT of an impulse response is called the frequency response (or transfer function) of an LTI system and is denoted by H(). e jw0 n jw H (e ) H (e jw0 ) × e jw0 n The output sequence is the input exponential sequence modified by the response of the system at frequency w0 Response to a complex exponential ∑ Ak e jwk n jw H (e ), h(n) ∑ k A k H ( e jwk ) e jwk n k In general, the frequency response H is a complex function of w. The magnitude |H| is called the magnitude (gain) response function, and the the angle is called the phase response function. Response to sinusoidal sequences x(n) = A cos( w0 n + θ 0 ) y (n) = A | H (e jw0 ) | cos( w0 n + θ 0 + ∠H (e jw0 )) x(n) = ∑ Ak cos( wk n + θ k ) k y (n) = ∑ Ak | H (e jw0 ) | cos( wk n + θ k + ∠H (e jw0 )) k Steady-state response Response to arbitrary sequences x(n) h(n) y ( n ) = h ( n) * x ( n ) jw X (e ) H (e jw ), h(n) Y (e jw ) = H (e jw ) X (e jw ) Condition: 1. Absolutely summable sequence 2. LTI system Frequency response function from difference equations When an LTI system is represented by the difference equation, y ( n) + ∑l =1 al y (n − l ) = ∑m =0 bm x(n − m) N M then to evaluate its frequency response , we would need the impulse response h(n). We know that when x(n) = e , then y(n) must be jwn H (e jw )e jwn ∑ − jwm M b e H (e jw )= m =0 m 1 + ∑l =0 al e − jwl N Implementation in Matlab H (e jw ) = b exp(− jmT w). / a exp(− jl T w) b = [b0 , b1 , LbM ] a = [a0 , a1 ,L a N ] m = [0,1,2,L , M ] l = [0,1,2, L , N ] k = [0,1,2,L , K ] w = pi * k / K Sampling & reconstruction of analog signals Analogy signals can be converted into discrete signals using sampling and quantization operations: analogy-to- digital conversion, or ADC Digital signals can be converted into analog signals using a reconstruction operation: digital-to-analogy conversion, or DAC Using Fourier analysis, we can describe the sampling operation from the frequency-domain view-point, analyze its effects and then address the reconstruction operation. We will also assume that the number of quantization levels is sufficiently large that the effect of quantization on discrete signals is negligible. Sampling Continuous-time Fourier transform and inverse CTFT +∞ X a ( jΩ) = ∫−∞ xa (t )e − jΩt dt 1 +∞ jΩt xa (t ) = ∫−∞ a X ( j Ω ) e dΩ 2π 1. Absolutely integrable 2. Omega is an analogy frequency in radians/sec Sampling Sample xa(t) at sampling interval Ts sec apart to obtain the discrete-time signal x(n) x(n) = xa (nTs ) 1 +∞   w 2π  X (e ) = jw Ts ∑ X a  j T − T l  l = −∞   s s  1. X is a countable sum of amplitude-scaled, frequency- scaled, and translated version of Xa 2. The above relation is known as the aliasing formula The analog and digital frequencies w = ΩTs 1 Fs = Fs: the sampling frequency, sam/sec Ts – Amplitude scaled factor: 1/Ts; – Frequency-scaled factor: ω=ΩTs (ω=0~2π) – Frequency-translated factor: 2πk/Ts; - Ω0TS Ω0TS - Ω0TS Ω0TS Sampling Interpretation Suppose signal band is limited to ±Ώ0, If Ts is small, Ώ0Tsπ, or F0= Ώ0/2 π > Fs/2=1/2Ts Then the freq. resp. of x(t) is a overlaped replica of its analog signal xa(t), so cannot be reconstructed Band-limited signal A signal is band-limited if there exists a finite radians frequency Ώ0 such that Xa(j Ώ) is zero for | Ώ |> Ώ0. The frequency F0= Ώ0 /2pi is called the signal bandwidth in Hz Referring to Fig.3.10, if pi> Ώ0Ts, then 1  w π w π X (e ) = X a  j ; − < ≤ jw Ts  Ts  Ts Ts Ts Sampling Principle A band-limited signal xa(t) with bandwidth F0 can be reconstructed from its sample values x(n)=xa(nTs) if the sampling frequency Fs=1/Ts is greater than twice the bandwidth F0 of xa(t) , Fs >2 F0. Otherwise aliasing would result in x(n). The sampling rate of 2 F0 for an analog band-limited signal is called the Nyquist rate. Reconstruction Impulse train Ideal lowpass filter x(n) conversion xa (t ) +∞ ∑ x(n)δ (t − nTs ) = L + x(−1)δ (t + Ts ) + x(0)δ (t ) + x(1)δ (t − Ts ) + L n = −∞ +∞ xa (t ) = ∑ x(n)sinc[ Fs (t − nTs )] Interpolating formula n = −∞ 1. Lowpass filter band-limited to the [-Fs/2,Fs/2] band 2. The ideal interpolation is not practically feasible because the entire system is noncausal and hence not realizable. Reconstruction Practical D/A converters Zero-order-hold (ZOH) interpolation: In this interpolation a given sample value is held for the sample interval until the next sample is received. It can be obtained by filtering the impulse train through an interpolating filter of the form xˆa (t ) = x(n), nTs ≤ n ≤ (n + 1)Ts 1 0 ≤ t ≤ Ts h0 (t ) =  0 otherwise Zero-order-hold (ZOH) interpolation The resulting signal is a piecewise-constant (staircase) waveform which requires an appropriately designed analog post-filter for accurate waveform reconstruction. x(n) ZOH xˆ a (t ) Post-Filter xa (t ) First-order-hold (FOH) interpolation In this case the adjacent samples are joined by straight lines.  t 1 + T 0 ≤ t ≤ Ts  s  t h1 (t ) = 1 − Ts ≤ t ≤ 2Ts  Ts  0 otherwise  Cubic-order-hold (COH) interpolation This approach uses spline interpolation for a smoother, but not necessarily more accurate, estimate of the analog signal between samples. Hence this interpolation does not require an analog post- filter The smoother reconstruction is obtained by using a set of piecewise continuous third-order polynomials called cubic splines xa (t ) = α 0 ( n) + α1 (n)(t − nTs ) + α 2 (n)(t − nTs ) 2 + α 3 (n)(t − nTs ) 3 , nTs ≤ t ≤ ( n + 1)Ts