# (6) Demonstrate aliasing by sampling a chirp

We demonstrate aliasing by sampling a linear chirp signal and then try to reconstruct the original signal using a cubic spline interpolator through the samples. Ideally, we should do this via the Shannon-Whittaker sinc function but alas not in GMT yet. As the frequency of the chirp increases we find it harder and harder to reconstruct a reasonable representation of the original signal from the samples. The morale is you need to sample data as often as you are able to. Here, we added a title slide visible for 6 seconds, then fade out to the animation. The scripts are a bit longer due to lots of little details. The finished movie is available in our YouTube channel as well: https://youtu.be/3vB53hoLsls The movie took ~3 minutes to render on a 24-core MacPro 2013.

    #!/usr/bin/env bash
    #
    # We demonstrate aliasing by sampling a linear chirp signal and then try to reconstruct
    # the original signal using a cubic spline interpolator through the samples.  Ideally, we
    # should do this via the Shannon-Whittaker sinc function but alas not in GMT yet.  As the
    # frequency of the chirp increases we find it harder and harder to reconstruct a reasonable
    # representation of the original signal from the samples.  The morale is you need to sample
    # data as often as you are able to.  Here, we added a title slide visible for 6 seconds, then
    # fade out to the animation.  The scripts are a bit longer due to lots of little details.
    # The finished movie is available in our YouTube channel as well:
    # https://youtu.be/3vB53hoLsls
    # The movie took ~3 minutes to render on a 24-core MacPro 2013.

    # 0. Initial parameters
    cat << 'EOF' > inc.sh
    R=-R-7.5/2.5/-1.5/2	# Fixed plot domain window
    J=-JX22c/11.5c		# Frame size after removing margin space
    f=2			# Frequency of chirp in Hz at end time
    rate=24			# Frames per seconds
    frames=$(gmt math -Q 60 $rate MUL =)
    EOF
    # 1. Make a title slide explaining things
    cat << 'EOF' > title.sh
    gmt begin
    	echo "12 11.5 Demonstration of aliasing by sampling a linear chirp" | gmt text -R0/24/0/13.5 -Jx1c -F+f26p,Helvetica-Bold+jCB -X0 -Y0
    	echo "12 10.5 y(t) = cos (2@~p@~t@+2@+/60)" | gmt text -F+f36p,Times-Italic+jTC
    	gmt text -M -F+f14p <<- END
    	> 12 6.5 16p 20c j
    	We will simulate sampling the continuous phenomenon described by @%6%y(t)@%% every 0.5 seconds,
    	meaning our Nyquist frequency is 1 Hz.  The samples are then interpolated with a cubic spline to
    	reconstruct the original signal@+*@+.  At first, this works great, but as the chirp increases
    	in frequency we find the interpolation starts to deviate from the actual phenomenon, and eventually
    	the reconstruction is clearly dominated by aliased frequencies much longer than those in the phenomenon.
    	END
    	gmt legend -Dx12c/3.5c+w20c+jTC+l1.2 -C0.3c -F+p+gazure1 <<- END
    	N 3
    	V 0 1p
    	S 1.5c - 2.5c - 1p,red 3.2c Chirp (phenomenon)
    	S 1.5c c 0.3c blue - 2c Sampled values
    	S 0.5c - 2.5c - 2.5p,blue 2.2c Spline interpolator
    	END
    	echo "@+*@+Sorry, we do not have a Shannon-Whittaker @%6%sinc@%% interpolator in GMT (yet)" | gmt text -F+f10p+cBR -Dj0.5c
    gmt end show
    EOF
    # 2. Create background plot and build data files needed in the main script
    cat << 'EOF' > pre.sh
    gmt begin
    	gmt plot $R $J -X1c -Y1c -B+gcornsilk -S1c -Ggreen <<- END
    	0	-1.5	t
    	0	+2.0	i
    	END
    	gmt basemap -Bxa2.5g10 -By0g10  --FORMAT_FLOAT_MAP=%+5.1f
    	# Display Nyquist frequency in lower left corner
    	echo "Nyquist frequency = 1 Hz" | gmt text -F+f18p,Helvetica-Bold+jBL+cBL -Dj0.3c
    gmt end
    # Build chirp data sets for 1 minute (60 secs); one every 1 ms and one every 0.5 sec as samples
    gmt math -T0/60/0.001 T 2 POW 2 DIV 60 DIV $f MUL 2 MUL PI MUL COS = chirp.txt
    gmt math -T0/60/0.5 -Ca T -C1 2 POW 2 DIV 60 DIV $f MUL 2 MUL PI MUL COS = chirp_samples.txt
    gmt math -T0/$frames/1 T $rate DIV = frame_times.txt
    EOF
    # 3. Set up the main frame script
    cat << 'EOF' > main.sh
    gmt begin
    	# Shift the chirp in time to simulate paper movement
    	gmt math chirp.txt -C0 ${MOVIE_COL1} SUB = chirp_shifted.txt
    	# Plot the shifted chirp
    	gmt plot $R $J chirp_shifted.txt -W1p,red -X1c -Y1c
    	# Compute index of most recent sample number
    	last_sample=$(gmt math -Q ${MOVIE_FRAME} $rate 2 DIV DIV FLOOR RINT =)
    	# Extract all the old samples before the present
    	gmt convert chirp_samples.txt -Z:$last_sample > tmp.txt
    	if [ -s tmp.txt ]; then
    		gmt math tmp.txt -C0 ${MOVIE_COL1} SUB = samples.txt
    		gmt plot -Sc0.3c -Gblue samples.txt
    	fi
    	# Take a new sample every 12 frames = 0.5 seconds
    	take_sample=$(gmt math -Q ${MOVIE_FRAME} $rate 2 DIV MOD 0 EQ =)
    	if [ ${MOVIE_FRAME} -gt 12 ]; then	# Interpolating up to most recent sample
    		gmt sample1d samples.txt -I0.001 > resampled.txt
    		gmt plot -W2.5p,blue resampled.txt
    	fi
    	if [ $take_sample -eq 1 ]; then	# Take and plot sample at zero time
    		y=$(gmt math -Q ${MOVIE_COL1} 2 POW 2 DIV 60 DIV $f MUL 2 MUL PI MUL COS =)
    		echo 0 $y | gmt plot -Sc0.5c -Gred
    	fi
    	# Add time counter in upper left corner
    	printf "t = %6.3f s\n" ${MOVIE_COL1} | gmt text -F+f18p,Helvetica-Bold+jTL+cTL -Dj0.3c
    	# Add cycles counter in upper right corner
    	fnow=$(gmt math -Q ${MOVIE_COL1} 60 DIV $f MUL =)
    	printf "f = %6.4f Hz\n" $fnow | gmt text -F+f16p,Helvetica-Bold+jTR+cTR -Dj0.3c
    	# Add frame counter in lower right corner
    	printf "%04d\n" ${MOVIE_FRAME} | gmt text -F+f14p,Helvetica-Bold+jBR+cBR -Dj0.3c
    gmt end
    EOF
    # 4. Run the movie
    source inc.sh
    gmt movie main.sh -Sbpre.sh -CHD -Iinc.sh -Tframe_times.txt -D$rate -Etitle.sh+d6s+fo1s+gwhite -Nanim06 -H8 -Zs -Fmp4 -W -V