Magnitude Plot Incorrect Frequency

Hi there, I'm having a problem plotting the magnitude spectrum of simple cos wave.

dt = .001;
t = -.5:dt:.5;
x1 = 2*cos(2*pi*20*t);
Wmax = 2*pi*100;
K = 2000;
w = (0:K-1) * Wmax / K;

X1 = x1 * exp(-j * t' * w) * dt;
mag_X1 = 2 * abs(X1);
display (X1);

subplot(2,1,1);
plot(t, x1);
xlabel('time');
ylabel('x(t)');
title('time domain');

subplot(2,1,2);
plot(w/2*pi, mag_X1);
axis([0 1000 0 7]);
xlabel('frequency Hz');
ylabel('|x(w)|');
title('magnitude spectrum');

The problem I'm having is that the plot shows a spike at 200 Hz rather than 20 Hz, if somebody could help me understand where I'm going wrong it would be greatly appreciated.

Answer 1

[Paul]

Recheck the conversion from rad/sec to Hz in the plot command for the magnitude of the spectrum

[Matt]

I'm not following? I'm under the impression that the frequency in hertz is simply the angular velocity divided by 2pi? Assuming it has something to do with wmax / k at the top?

[Paul]

Yes, the relationship is. Is that what the code implements?

Suppose omega = 6.28 so that f is approximately equal to 1. The code in the question would compute f as

w = 6.28;
f = w/2*pi

f = 9.8646

which is incorrect. Recall that Matlab follows PEMDAS for the basic math operators (Operator Precedence), so that computation for f is effectively being evaluated as

f = (w/2)*pi

f = 9.8646

[Matt]

Thanks so much, can't believe how silly I am.

Answer 2

[Hassaan]

% Time step
dt = 0.001;

% Time vector
t = -0.5 : dt : 0.5;

% Define the signal: x1 = 2 cos(2π·20t)
x1 = 2 * cos(2*pi*20*t);

% Maximum angular frequency (rad/s)
Wmax = 2*pi*100;

% Number of frequency samples
K = 2000;

% Frequency vector in rad/s
w = (0 : K-1) * (Wmax / K);

% Compute Fourier transform via integration (numerical)
X1 = x1 * exp(-1j * t' * w) * dt;

% Magnitude (scaled by 2 for plotting)
mag_X1 = 2 * abs(X1);

% --- Plot time-domain signal ---
figure;
subplot(2,1,1);
plot(t, x1);
xlabel('Time (s)');
ylabel('x(t)');
title('Time Domain');

% --- Plot magnitude spectrum ---
subplot(2,1,2);
% Use w/(2*pi) to convert rad/s to Hz
plot(w/(2*pi), mag_X1);
xlabel('Frequency (Hz)');
ylabel('|X(\omega)|');
title('Magnitude Spectrum');

% Optional axis limits for clarity
axis([0 100 0 7]);

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