Brief Notes on 1st Part of ME579

Would like: X(f ), the Fourier Transform of x(t), or Ck, the Fourier Series coefficients, if signal is periodic. We have: xn=x(nΔ) samples of x(t) every Δ seconds for n = 0,1,2,3….N-1. T=NΔ seconds is the duration of the signal. And can do Discrete Fourier Transforms (DFT) to give: Xk for k=0,1,2,….N-1 with corresponding frequencies fk = 0, fs/N, 2fs/N, ….(N-1)fs/N.

What is the relationship between Xk and X(f )? For Periodic Signals:

What is the relationship between Xk and Ck?

Fourier Transforms

X( f )= x( t )e− j2π ft dt−∞

∞

∫ x-units/Hz

x( t )= X( f )e+ j2π ft df−∞

∞

∫

Both continuous functions Both involve infinite integrals Difficult to do these on a computer for any function Your measurements will be samples of signalsNeed to sample signals and spectra to do calculations. Need to sample to plot functions on the computer: pick some values of t à tn n=0,1, … and evaluate x(tn) pick some values of f à fk k=0,1,2,… and evaluate X(f)

Complex Form of Fourier Series (Periodic Signals)

Ck=1T

x( t )e− j2π k

Ttdt

0

T∫ x-units, T = period,

fk=kT

Hz

x( t )= Ck

k=−∞

∞

∑ e+ j2π k

Tt

Already discrete in frequency Continuous in time, but only integrating over 1 period

Relationship between Complex Form of Fourier Series and Sines and Cosines Form

Ck=1T

x( t )cos 2π kTt

⎛

⎝⎜

⎞

⎠⎟dt

0

T∫ − j 1

Tx( t )sin 2π k

Tt

⎛

⎝⎜

⎞

⎠⎟dt

0

T∫

=Ak2− jBk2

C−k=C*

k=Ak2+ jBk2

, because the signal is real.

x( t )=C0 + Cke+ j2π k

Tt+C

−ke− j2π k

Tt

k=1

∞

∑

=A02+ A

kk=1

∞

∑ cos 2π kTt

⎛

⎝⎜

⎞

⎠⎟+ Bk sin 2π

kTt

⎛

⎝⎜

⎞

⎠⎟

Fourier Transform to Discrete Fourier Transform

1. WINDOWED MADE LENGTH FINITE à SPECTRAL LEAKAGE Product in Time Convolution in Frequency (Smoothed Spectrum) 2. SAMPLED IN TIME ALIASING [ADC: CLIPPING, QUANTIZATION]

Multiplied by a delta train Scaled & Periodic in Frequency Calculation 3. SAMPLED IN FREQUENCY ALIASING

xw( t )= x( t ).w( t ) X

w( f )= X( g )W( f − g )dg∫

Xsf( fk)= X( f )δ( f − k.df )∑ → x

sf( t )= 1

dfx( t − rT )

r=−∞

∞

∑ =T x( t − rT )r=−∞

∞

∑

xs( t )= δ( t − nΔ )x( t )∑ X

s( f )= 1

ΔX( f − qf

s)= x( nΔ )e− j2π f Δnn=−∞

∞

∑q=−∞

∞

∑

Multiplied by a delta train in Frequency Scaled & Periodic in Time

Brief Notes on 1st Part of ME579

Discrete Fourier Transform:Signal sampled and finite in time 0àT=NΔ, fs = 1/Δ, tn=n/fs=nΔ

Spectrum sampled in frequency 0à fs, df=1/T, fk=k/T=k.fs/N

à  periodic in time and frequency

à windowing in time smooths out spectrum of the sampled signal

à  wrap around effects in both time and frequency (two forms of aliasing)

Discrete Fourier Transform (DFT)

Xk= x

ne− j2π nk

N

n=0

N−1∑ , k = 0,1,2,...N −1; f

k= k. fs

N Hz.

xn=

1N

Xke+ j2π nk

N

k=0

N−1∑ , n = 0,1,2,...N −1; t

n= nΔ secs.

Inverse Discrete Fourier Transform (IDFT)

(Inverse) Fast Fourier Transform - (I)FFT A clever way of calculating the Forward and Inverse Discrete Fourier Transforms that is fast if N is highly factorisable into high powers of prime integers: e.g., if N is a power of 2, e.g., 128, 1024, 8192, …. Ratio of Operations Needed (approximately): Direct DFT Formula

Radix 2 FFT = N

log2N

N Radix 2 FFT Direct Ratio (times faster)

8 24 64 2.67 128 896 16,384 18.3 1024 10,240 1,048,576 102.4 8192 106,496 67,108,864 640.1

Domains: Time and Frequency

Time Domain Frequency Domain Sampled/Discrete Periodic

Multiplication Convolution

Even Function Real Odd Function Imaginary

Real Signal Complex Conjugate

Symmetry C-k = Ck

* X(-f) = X(f)*

Narrow in Time Broad in Frequency Broad in Time Narrow in Frequency

When do we Zero Pad Signals?

1.  When result of an operation should yield a longer signal than original signal(s). e.g. convolution and time-delay.

2.  When we want to have a clearer picture of Xs(f), the Fourier Transform of the sampled signal, xs(t). [Would prefer to transform more data to get better resolution, i.e., a longer duration window w(t) which means a spikier, more localized W(f). Use zero padding when we don’t have any more data to transform.]

Discrete Fourier Transform (DFT), Xk Relationship to X(f) and Ck

Assume no aliasing when sampling i.e., fs > 2 fmax

Have x(nΔ) for n=0,1,2,…N-1; Xk = D.F.T.(xn); k=0,1,2,3,…N-1.

Periodic Signals: NΔ = a whole number of periods = q Tp

Ck = Xk/N for -(N/2) < k < (N/2) Transients: (some aliasing will occur)

X(f) ≈ Δ Xk for -fs/2 < f < fs/2

Discrete Fourier Transform (DFT), Xk

Symmetry about 0 and fs/2 Real Part and Magnitude: even symmetry about these points Imaginary Part and Phase: odd symmetry about these points Periodic:

Xk for k=+(N/2),+(N/2)+1, ……. N-1 equal to Xk for k=-(N/2),-(N/2)+1, ……. –1

fftshift in Matlab will rearrange the DFT for you, but you have to make the corresponding frequency vector: fk=(-N/1:(N/2)-1)/Tp Hz.

Analog to Digital - Digital to Analog

ADC Characteristics: Input/Output Range, No. of Bits, fs, ADC Type

Analog to Digital Conversion (ADC):

Quantization Error, Clipping, Sample and hold, Aliasing Anti-aliasing Filters and Sample Rate.

fhighest = highest frequency of interest fc = filter cut-off frequency fmax = highest frequency in filtered signal

Sample Rate = fs > 2 fmax

fhighest < fc < fmax < fs/2

Analog to Digital - Digital to AnalogDigital to Analog Conversion: Codes to Signal Similar issues as for ADC Zero-order hold characteristics Sinc function in frequency: Δ sinc(πfΔ) . exp(-jπfΔ) with a time delay of half one sample. Distorts signal in range -fmax< f <fmax, Distortion less if fs >>> fmax. Can pre-compensate for this delay by designing an appropriate digital filter, and pre-filtering signal. Reconstruction Filter: fmax < fc << fs/2. Similar characteristics to anti-aliasing filter in ADC.

Other Things We Have Looked At

We use the delta function, δ(t) or δ(f), in a lot of our theory and proofs. •  Sifting property in integrals •  Integral from -∞ to + ∞ of exp( ±j2π f t ) •  Fourier Transforms of periodic signals: sines and cosines •  Sampling theory – delta trains

We looked at the FFT algorithm (not on exam), which is a computationally efficient way of calculating a DFT Convolution of continuous and discrete signals: signals passing through systems.

All the Transforms: timeßàfrequency•  Complex Fourier Series x(t) ⇔ Ck

periodic in time, discrete in frequency

•  Fourier Transforms x(t) ⇔ X(f) continuous in time and frequency

•  Fourier Transform of a sampled signal: xs(t) ⇔ Xs(f) OR x(nΔ) ⇔ Xs(f)

Xs(f) = (1/Δ) Σq X(f - q fs) discrete in time, periodic and continuous in frequency

•  Discrete Fourier Transform (finite set of data used)

xn ⇔ Xk; n and k: 0,1,2…..N-1. periodic and discrete in both time and frequency

Using the Discrete Fourier TransformFourier Series coefficients (no aliasing and N corresponds to exactly a whole no. of periods)

Ck = Xk/N for k=0,1,2…N/2.

Approximate the Fourier Transform of x(t) à X(f) at frequencies: f = k.fs/N (if signal is of finite length and there is no aliasing – usually not the case)

X(f) at f = k.fs/N ≈ Δ Xk for k=0,1,2…(N/2). Sampled version of Xs(f) (sampled signal, xs(t) was of finite length = N points)

Xs(f) at f = k.fs/N = Xk Can zero pad time sequence to evaluate spectrum at more frequency points. Does not increase true resolution but gives you a better picture of the underlying Xs(f).

Using the Discrete Fourier TransformConvolution of 2 discrete signals via the frequency domain. xn is Nx points long hn is Nh points long yn = hn * xn is Nx + Nh -1 points long. Find N = 2q ≥ Nx + Nh -1.

yn = first Nx + Nh -1 points of REAL{ IDFT[ DFT[xn , N] . DFT[hn , N] ] } point by point multiplication yn start = xn start time + hn start time tyn = yn start + n.Δ, n=0,1,2,….. Nx + Nh