0.2 Preliminaries

Fourier transform (ft)

Definition:

Properties:

• Linearity: $\mathcal{F}\{c_1{w}_1\left(t\right)+c_2{w}_2\left(t\right)\}=c_1{W}_1\left(f\right)+c_2{W}_2\left(f\right)$ .
• Real-valued $w\left(t\right)\Rightarrow \{\begin{array}{c}\text{conjugate symmetric}W\left(f\right)\\ \left|W\right(f\left)\right|\text{symmetric around}f=0\end{array}$ .

“bandwidth”

Dirac delta (or “continuous impulse”) $\delta (·)$

An infinitely tall and thin waveform with unit area :

that's often used to “kick” a system and see how it responds.

Key properties:

1. Sifting: ${\displaystyle {\int }_{-\infty }^{\infty }w\left(t\right)\delta (t-q)dt=w\left(q\right)}$ .
2. Time-domain impulse $\delta \left(t\right)$ has a flat spectrum:
   $$\mathcal{F}\left\{\delta \left(t\right)\right\}={\int }_{-\infty }^{\infty }\delta \left(t\right)e^{-j2\pi ft}dt=1\text{(for}\text{all}f\text{)}.$$
3. Freq-domain impulse $\delta \left(f\right)$ corresponds to a DC waveform:
   $${\mathcal{F}}^{-1}\left\{\delta \left(f\right)\right\}={\int }_{-\infty }^{\infty }\delta \left(f\right)e^{j2\pi ft}df=1\text{(for}\text{all}t\text{)}.$$

Frequency-domain representation of sinusoids

Notice from the sifting property that

Thus, Euler's equations

and the Fourier transform pair $e^{j2\pi f_{o}t}\leftrightarrow \delta (f-f_o)$ imply that

Often we draw this as

Frequency domain via matlab

Fourier transform requires evaluation of an integral. What do we do if we can't define/solve the integral?

1. Generate (rate- $\frac{1}{T_s}$ ) sampled signal in MATLAB.
2. Plot magnitude of Discrete Fourier Transform (DFT) using plottf.m (from course webpage).

Square-wave example:

f = 10; t_max = 2;Ts = 1/1000; t = 0:Ts:t_max;x = sign(cos(2*pi*f*t)); plottf(x,Ts);

Noise-wave example

t_max = 1; Ts = 1/1000;x = randn(1,t_max/Ts); plottf(x,Ts);

Notice that plottf.m only plots frequencies $f\in [-\frac{1}{2T_s},\frac{1}{2T_s})$ .

Linear time-invariant (lti) systems

An LTI system can be described by either its “impulse response” $h\left(t\right)$ or its “frequency response” $H\left(f\right)=\mathcal{F}\left\{h\right(t\left)\right\}$ .

Input/output relationships:

• Time-domain: Convolution with impulse response $h\left(t\right)$

  

• Freq-domain: Multiplication with freq response $H\left(f\right)$

  

Linear filtering

Freq-domain illustration of LPF, BPF, and HPF:

Lowpass filters

Ideal non-causal LPF (using $sinc\left(x\right):=\frac{sin\left(\pi x\right)}{\pi x}$ ):

Ideal LPF with group-delay t _o :

A causal linear-phase LPF with group-delay t _o :

but MATLAB can give better causal linear-phase LPFs...

In MATLAB, generate $\frac{1}{T_s}$ -sampled LPF impulse response via

h = firls(Lf, [0,fp,fs,1], [G,G,0,0])/Ts;

where...

The commands firpm and fir2 have the same interface, but yield slightly different results (often worse for our apps).

In MATLAB, perform filtering on $\frac{1}{T_s}$ -sampled signal x via

y = Ts*filter(h,1,x);   or   y = Ts*conv(h,x);

t_max = 3; Ts = 1/1000; x = randn(1,t_max/Ts);h = firls(100,[0,0.2,0.4,1],[1,1,0,0])/Ts; y = Ts*filter(h,1,x);subplot(3,1,1); plottf(x,Ts,’f’);ylabel(’|X(f)|’) subplot(3,1,2);plottf(h,Ts,’f’); ylabel(’|H(f)|’)subplot(3,1,3); plottf(y,Ts,’f’);ylabel(’|Y(f)|’)