Fix bug in FFT

The FFT routine does not work for odd N
Solution is to add DFT and use it when N is odd

main.cpp

@@ -1909,8 +1909,31 @@ whisper_vocab::id whisper_sample_timestamp(
     return probs_id[0].second;
 }
 
+// naive Discrete Fourier Transform
+// input is real-valued
+// output is complex-valued
+void dft(const std::vector<float> & in, std::vector<float> & out) {
+    int N = in.size();
+
+    out.resize(N*2);
+
+    for (int k = 0; k < N; k++) {
+        float re = 0;
+        float im = 0;
+
+        for (int n = 0; n < N; n++) {
+            float angle = 2*M_PI*k*n/N;
+            re += in[n]*cos(angle);
+            im -= in[n]*sin(angle);
+        }
+
+        out[k*2 + 0] = re;
+        out[k*2 + 1] = im;
+    }
+}
+
 // Cooley-Tukey FFT
-// poor man's implmentation - use something better
+// poor man's implementation - use something better
 // input is real-valued
 // output is complex-valued
 void fft(const std::vector<float> & in, std::vector<float> & out) {
@@ -1924,6 +1947,11 @@ void fft(const std::vector<float> & in, std::vector<float> & out) {
         return;
     }
 
+    if (N%2 == 1) {
+        dft(in, out);
+        return;
+    }
+
     std::vector<float> even;
     std::vector<float> odd;
 
@@ -2014,9 +2042,20 @@ bool log_mel_spectrogram(
                 // FFT -> mag^2
                 fft(fft_in, fft_out);
 
-                for (int j = 0; j < n_fft; j++) {
+                for (int j = 0; j < fft_size; j++) {
                     fft_out[j] = (fft_out[2*j + 0]*fft_out[2*j + 0] + fft_out[2*j + 1]*fft_out[2*j + 1]);
                 }
+                for (int j = 1; j < fft_size/2; j++) {
+                    //if (i == 0) {
+                    //    printf("%d: %f %f\n", j, fft_out[j], fft_out[fft_size - j]);
+                    //}
+                    fft_out[j] += fft_out[fft_size - j];
+                }
+                if (i == 0) {
+                    //for (int j = 0; j < fft_size; j++) {
+                    //    printf("%d: %e\n", j, fft_out[j]);
+                    //}
+                }
 
                 // mel spectrogram
                 for (int j = 0; j < mel.n_mel; j++) {
@@ -2048,6 +2087,7 @@ bool log_mel_spectrogram(
             mmax = mel.data[i];
         }
     }
+    //printf("%s: max = %f\n", __func__, mmax);
 
     mmax -= 8.0;