Absolute Magnitude of the FFT of Sinusoids with Unit Amplitudes and Unit Periods

Vladimir Kulyukin

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Problem

This post briefly describes how to obtain and plot the absolute magnitude of a sinusoid with a unity amplitude and period. and normalized frequencies of the FFT. Figure 1 sketches a proof of why the absolute magnitude of the FFT of length N of  a sinusoid with a unit amplitude and period  is equal to N/2.
  

Figure 12. Plots of sinusoid 04, its absolute magnitudes, and normalized frequencies

Figure 12. Plots of sinusoid 04, its absolute magnitudes, and normalized frequencies

Figure 1. Absolute Magnitude of the FFT

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Sinusoid 01
 

Figure 2.  Sinusoid 01 with A=1, T=1, L=4, x=1

Here is the MATLAB code that encodes sinusoid 01 defined in Figure 2.

N=64;       %% this should be a power of 2
T=1;        %% this is the period
A=1;        %% amplitude
phi=0.0;    %% phase of the sinusoid
fs = 1/T;   %% fs
t = 0:N-1   %% discrete time axis
L = 4;
x = 1;
wx = x*2*pi*fs/L %% wx frequency
s_01a = A*cos(wx*t + phi); %% input signal
FFT_01a = fft(s_01a); %% spectrum
%% Plot the sinusoid
figure(1);
subplot(3,1,1);
plot(t, s_01a, '*k');
ti = 0:.1:N-1; %% interpolate time axis in 0.1 increments
hold on;
plot(ti, A*cos(wx*ti + phi), '-k'); grid off;
title('Sinusoid signal: Example 01a');
xlabel('Time (samples)');
ylabel('Amplitude');
text(-8, 1, 'a)');
hold off;
%% Plot absolute magnitude at normalized frequencies
magFFT_01a = abs(FFT_01a);
disp(magFFT_01a);
tn = 0:1/N:1-1/N;
subplot(3, 1, 2);
stem(tn, magFFT_01a, 'ok'); grid on;
xlabel('Frequency (cycles per sample)');
ylabel('Magnitude (linear)');
text(-.11, 40, 'b');
%% Plotting dB scales
dbSpectrum_01a = 20*log10(magFFT_01a); %% convert into dB
subplot(3, 1, 3);
plot(tn, dbSpectrum_01a, '--ok'); grid on;
axis([0 1 -350 50]);
xlabel('Normalized frequency (cycles per sample)');
ylabel('Magnitude (dB)');
text(-.15, 50, 'c)');


Figure 3 gives the graphs computed and plotted by the above MATLAB code.

Figure 3. Plots of sinusoid 01, its absolute magnitudes, and normalized frequencies

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Sinusoid 02

Figure 7. Sinusoid 02 with A=1, T=1, L=8, x=1

Here is the MATLAB code for sinusoid 02 defined in Figure 7.

N=64;       %% this should be a power of 2
T=1;        %% this is the period
A=1;        %% amplitude
phi=0.0;    %% phase of the sinusoid
fs = 1/T;   %% fs
t = 0:N-1   %% discrete time axis
L = 8;
x = 1;
wx = x*2*pi*fs/L %% wx frequency
s_01b = A*cos(wx*t + phi); %% input signal
FFT_01b = fft(s_01b); %% spectrum
%% Plot the sinusoid
figure(1);
subplot(3,1,1);
plot(t, s_01b, '*k');
ti = 0:.1:N-1; %% interpolate time axis in 0.1 increments
hold on;
plot(ti, A*cos(wx*ti + phi), '-k'); grid off;
title('Sinusoid signal: Example 02');
xlabel('Time (samples)');
ylabel('Amplitude');
text(-8, 1, 'a)');
hold off;
%% Plot absolute magnitude at normalized frequencies
magFFT_01b = abs(FFT_01b);
disp(magFFT_01b);
tn = 0:1/N:1-1/N;
subplot(3, 1, 2);
stem(tn, magFFT_01b, 'ok'); grid on;
xlabel('Frequency (cycles per sample)');
ylabel('Magnitude (linear)');
text(-.11, 40, 'b');
%% Plotting dB scales
dbSpectrum_01b = 20*log10(magFFT_01b); %% convert into dB
subplot(3, 1, 3);
plot(tn, dbSpectrum_01b, '--ok'); grid on;
axis([0 1 -350 50]);
xlabel('Normalized frequency (cycles per sample)');
ylabel('Magnitude (dB)');
text(-.15, 50, 'c)');

Figure 8 gives the graphs computed and plotted by the above MATLAB code.

Figure 8. Plots of sinusoid 02, its absolute magnitudes, and normalized frequencies

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Sinusoid 03
 

Figure 9. Sinusoid with A=1, T=1, L=16, x=1

Here is the MATLAB code for sinusoid 03 in Figure 9.

N=64;       %% this should be a power of 2
T=1;        %% this is the period
A=1;        %% amplitude
phi=0.0;    %% phase of the sinusoid
fs = 1/T;   %% fs
t = 0:N-1   %% discrete time axis
L = 16;
x = 1;
wx = x*2*pi*fs/L %% wx frequency
s_01c = A*cos(wx*t + phi); %% input signal
FFT_01c = fft(s_01c); %% spectrum
%% Plot the sinusoid
figure(1);
subplot(3,1,1);
plot(t, s_01c, '*k');
ti = 0:.1:N-1; %% interpolate time axis in 0.1 increments
hold on;
plot(ti, A*cos(wx*ti + phi), '-k'); grid off;
title('Sinusoid signal: Example 03');
xlabel('Time (samples)');
ylabel('Amplitude');
text(-8, 1, 'a)');
hold off;
%% Plot absolute magnitude at normalized frequencies
magFFT_01c = abs(FFT_01c);
disp(magFFT_01c);
tn = 0:1/N:1-1/N;
subplot(3, 1, 2);
stem(tn, magFFT_01c, 'ok'); grid on;
xlabel('Frequency (cycles per sample)');
ylabel('Magnitude (linear)');
text(-.11, 40, 'b');
%% Plotting dB scales
dbSpectrum_01c = 20*log10(magFFT_01c); %% convert into dB
subplot(3, 1, 3);
plot(tn, dbSpectrum_01c, '--ok'); grid on;
axis([0 1 -350 50]);
xlabel('Normalized frequency (cycles per sample)');
ylabel('Magnitude (dB)');
text(-.15, 50, 'c)');

Figure 10 gives the graphs computed and plotted by the above MATLAB code.
 

Figure 10. Plots of sinusoid 03, its absolute magnitudes, and normalized frequencies

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Sinusoid 04
 

Figure 11. Sinusoid 04 with A=1, T=1, L=4, x=2

Here is the MATLAB source for  sinusoid 04 defined in Figure 11.

N=64;       %% this should be a power of 2
T=1;        %% this is the period
A=1;        %% amplitude
phi=0.0;    %% phase of the sinusoid
fs = 1/T;   %% fs
t = 0:N-1   %% discrete time axis
L = 4;
x = 2;
wx = x*2*pi*fs/L %% wx frequency
s_01d = A*cos(wx*t+phi); %% input signal
FFT_01d = fft(s_01d); %% spectrum
%% Plot the sinusoid
figure(1);
subplot(3,1,1);
plot(t, s_01d, '*k');
ti = 0:.1:N-1; %% interpolate time axis in 0.1 increments
hold on;
plot(ti, A*cos(wx*ti + phi), '-k'); grid off;
title('Sinusoid input: Example 04');
xlabel('Time (samples)');
ylabel('Amplitude');
text(-8, 1, 'a)');
hold off;
%% Plot absolute magnitude at normalized frequencies
magFFT_01d = abs(FFT_01d);
disp(magFFT_01d);
tn = 0:1/N:1-1/N;
subplot(3, 1, 2);
stem(tn, magFFT_01d, 'ok'); grid on;
xlabel('Frequency (cycles per sample)');
ylabel('Magnitude (linear)');
text(-.11, 40, 'b');
%% Plotting dB scales
dbSpectrum_01d = 20*log10(magFFT_01d); %% convert into dB
subplot(3, 1, 3);
plot(tn, dbSpectrum_01d, '--ok'); grid on;
axis([0 1 -350 50]);
xlabel('Normalized frequency (cycles per sample)');
ylabel('Magnitude (dB)');
text(-.15, 50, 'c)');

Figure 12 gives the graphs computed and plotted by the above MATLAB code.

Figure 12. Plots of sinusoid 04, its absolute magnitudes, and normalized frequencies

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