### Lecture 18 

# --- Examples of norm of FFT (periodogram)

# constant

ft <- fft(rep(1,8))

# boxcar

n  <- 64
t  <- 0:(n-1)

f1 <- fft(y1 <- c(rep(1,n/2),rep(0,n/2))); f1; plot(Mod(f1))
f2 <- fft(y2 <- c(rep(1,n/4),rep(0,n/2),rep(1,n/4)));f2; lines(Mod(f2))

Mod(f1)
Mod(f2)

Arg(f1)
Arg(f2)

# sinusoid

n  <- 64
t  <- 0:(n-1)
la <- 10/n

f1 <- fft(y1 <- sin(2*pi*t*la)); Mod(f1)
f1 <- fft(y1 <- sin(2*pi*(t+24)*la)); Mod(f1)
plot(Mod(f1))

xi <- la + 1/(5*n)
f2 <- fft(sin(2*pi*t*xi)); Mod(f2); lines(Mod(f2), col="lightblue")

xi <- la + 1/(3*n)
f3 <- fft(sin(2*pi*t*xi)); Mod(f3); lines(Mod(f3), col="pink")

xi <- la + 1/(2*n)
f4 <- fft(sin(2*pi*t*xi)); Mod(f4); lines(Mod(f4), col="black")

# sinusoid + outlier

n  <- 64
t  <- 0:(n-1)
la <- 10/n

y <- sin(2*pi*t*la)

outlier <- rep(0,n); outlier[6] <- 1

plot(y+outlier)

f <- fft(y + outlier ); points(Mod(f))

f1 <- fft(y); plot(Mod(f1))
f2 <- fft(outlier); plot(Mod(f2))



# --- Examples of norm of DFT (periodogram)