Lecture on Continuous and Discrete Fourier Transforms www.AssignmentPoint.com www.assignmentpoint.com

TOPICS 1. Infinite Fourier Transform (FT) 2. FT & generalised impulse 3. Uncertainty principle 4. Discrete Time Fourier Transform (DTFT) 5. Discrete Fourier Transform (DFT) 6. Comparing signal by DFS, DTFT & DFT 7. DFT leakage & coherent sampling www.assignmentpoint.com

Fourier analysis - tools Input Time Signal Frequency spectrum Periodic (period T) Discrete Continuous FT Aperiodic FS Note: j =-1,  = 2/T, s[n]=s(tn), N = No. of samples Discrete DFS Periodic (period T) Continuous DTFT Aperiodic DFT www.assignmentpoint.com

Fourier Integral (FI) Fourier Integral Theorem Any aperiodic signal s(t) can be expressed as a Fourier integral if s(t) piecewise smooth(1) in any finite interval (-L,L) and absolute integrable(2). Fourier Integral Theorem (3) Fourier analysis tools for aperiodic signals. (2) s(t) continuous, s’(t) monotonic (1) (3) Fourier Transform (Pair) - FT analysis synthesis Complex form Real-to-complex link www.assignmentpoint.com

Let’s summarise a little Signal  Periodic Aperiodic Domain  FS FI Time real ak, bk A(), B() Frequency FT complex ck C() www.assignmentpoint.com

Note: |ak|2 a0 as k0  2 a0 is plotted at k=0 From FS to FT FS moves to FT as period T increases: continuous spectrum T = 0.05 Pulse train, width 2 t = 0.025 T = 0.1 Frequency spacing 0 ! T = 0.2 FT Note: |ak|2 a0 as k0  2 a0 is plotted at k=0 www.assignmentpoint.com

f = /(2) = 1/T frequency spacing Getting FT from FS 2 FS defined f = /(2) = 1/T frequency spacing FT defined As f 0 , replace f , K , by df, 2f, 1 www.assignmentpoint.com

FT & Dirac’s Delta Dirac’s  defined The FT of the generalised impulse  (Dirac) is a complex exponential Hence FT of Dirac’s  property FT of an infinite train of : a.k.a. Sampling function, Shah(T) = Щ(T) or “comb” Note:  & Щ = “generalised “functions www.assignmentpoint.com

FT properties Time Frequency Linearity a·s(t) + b·u(t) a·S(f)+b·U(f) Multiplication s(t)·u(t) Convolution S(f)·U(f) Time shifting Frequency shifting Time reversal s(-t) S(-f) Differentiation j2f S(f) Parseval’s identity  h(t) g*(t) dt =  H(f) G*(f) df Integration S(f)/(j2f ) Energy & Parseval’s (E is t-to-f invariant) www.assignmentpoint.com

FT - uncertainty principle For effective duration t & bandwidth f   > 0 t•f   uncertainty product Bandwidth Theorem Fourier uncertainty principle For Energy Signals: E= |s(t)|2dt = |S(f)|2df <  Define mean values Define std. dev.  t•f  1/4 Implications • Limited accuracy on simultaneous observation of s(t) & S(f). • Good time resolution (small t) requires large bandwidth f & vice-versa. www.assignmentpoint.com

FT - example FT of 2-wide square window Fourier uncertainty Choose S(f) = 2 sMAX sync(2f) FT of 2-wide square window Choose t = |s(t)/s(0) dt| = 2, f = |S(f)/S(0) df|=1/(2) = half distance btwn first 2 zeroes (f1,-1 = 1/2) of S(f) then: t · f = 1 Fourier uncertainty Power Spectral Density (PSD) vs. frequency f plot. Note: Phases unimportant! www.assignmentpoint.com

Phone signals PSD masks FT - power spectrum POTS = Voice/Fax/modem Phone HPNA = Home Phone Network Phone signals PSD masks US = Upstream DS = Downstream From power spectrum we can deduce if signals coexist without interfering! Power Spectral Density, PSD(f) = dE/df = |S(f)|2 www.assignmentpoint.com

FT of main waveforms www.assignmentpoint.com

Discrete Time FT (DTFT) DTFT defined as: Note: continuous frequency domain! (frequency density function) analysis Holds for aperiodic signals synthesis Obtained from DFS as N   www.assignmentpoint.com

DTFT - convolution Digital Linear Time Invariant system: obeys superposition principle. h[t] = impulse response Convolution x[n] h[n] X(f) H(f) Y(f) = X(f) · H(f) www.assignmentpoint.com

DTFT - Sampling/convolution Time Frequency s[n] * u[n]  S(f) · U(f) , s[n] · u[n]  S(f) * U(f) (From FT properties) Sampling s(t) Multiply s(t) by Shah = Щ(t) www.assignmentpoint.com

Discrete FT (DFT) DFT defined as: ~ Frequency resolution synthesis analysis Frequency resolution Analysis frequencies fm DFT bins located @ analysis frequencies fm DFT ~ bandpass filters centred @ fm Applies to discrete time and frequency signals. Same form of DFS but for aperiodic signals: signal treated as periodic for computational purpose only. Note: ck+N = ck  spectrum has period N ~ www.assignmentpoint.com

DFT - pulse & sinewave a) rectangular pulse, width N ~ ck = (1/N) e-jk(N-1)/N sin(k)/ sin(k/N) ~ a) rectangular pulse, width N r[n] = 1 , if 0nN-1 0 , otherwise b) real sinewave, frequency f0 = L/N cs[n] = cos(j2f0n) ck = (1/N) ej{(Nf0-k)-(Nf0 -k)/N} (½) sin{(Nf0-k)}/ sin{(Nf0-k)/N)} + (1/N) ej{(Nf0+k)-(Nf0+k)/N} (½) sin{(Nf0+k)}/ sin{(Nf0+k)/N)} ~ i.e. L complete cycles in N sampled points www.assignmentpoint.com

DFT Examples DFT plots are sampled version of windowed DTFT www.assignmentpoint.com

DFT properties Time Frequency Linearity a·s[n] + b·u[n] a·S(k)+b·U(k) Multiplication s[n] ·u[n] Convolution S(k)·U(k) Time shifting s[n - m] Frequency shifting S(k - h) www.assignmentpoint.com

DTFT vs. DFT vs. DFS (b) (a) (d) (c) (f) (e) DTFT transform magnitude. Aperiodic discrete signal. (d) DFS coefficients = samples of (b). (c) Periodic version of (a). (f) DFT estimates a single period of (d). (e) Inverse DFT estimates a single period of s[n] www.assignmentpoint.com

DFT – leakage * * N·Magnitude Leakage (a) (b) (c) Spectral components belonging to frequencies between two successive frequency bins propagate to all bins. Leakage Ex: 32-bins DFT of 1 VP sinusoid sampled @ 32kHz. 1 kHz frequency resolution. (a) 8 kHz sinusoid * N·Magnitude * (b) 8.5 kHz sinusoid (c) 8.75 kHz sinusoid www.assignmentpoint.com

DFT - leakage example s(t) FT{s(t)} 0.25 Hz Cosine wave 5. Sampled windowed wave 1. Cosine wave 3. Windowed cos wave s[n] · u[n]  S(f) * U(f) (Convolution) 4. Sampling function 2. Rectangular window 1. Cosine wave Leakage caused by sampling for a non-integer number of periods www.assignmentpoint.com

DFT - coherent sampling s(t) FT{s(t)} 0.2 Hz Cosine wave 5. Sampled windowed wave 3. Windowed cos wave 1. Cosine wave s[n] ·u[n]  S(f) * U(f) (Convolution) 2. Rectangular window 1. Cosine wave 4. Sampling function Coherent sampling: NC input cycles exactly into NS = NC (fS/fIN) sampled points. www.assignmentpoint.com

DFT – leakage notes Leakage Affects Real & Imaginary DFT parts magnitude & phase. Has same effect on harmonics as on fundamental frequency. Affects differently harmonically un-related frequency components of same signal (ex: vibration studies). Leakage depends on the form of the window (so far only rectangular window). After coffee we’ll see how to take advantage of different windows. www.assignmentpoint.com