interpolation: why I can see the original sequence in the interpolated data only when (N/M) = integer

Question

I am trying to understand the zero-padding in frequency domain and discovered that original sequence in interpolated data only when (N/M) = integer where N and M are the lengths of output and input.

clear all;
close all;
M  = 9;
N  = 18;
m_p = (M - 1) / 2;
x1 = [2 4 5 2 1 3 1 7 9];
y1 = fft(x1);
y2 = (N/M) * [y1(1:m_p+1) zeros(1, N-M) y1(m_p+2:M)];
z_c = N - M;
z_f = 1 + (z_c-1) / 2
z_b = z_f - 1;
y3 = (N/M) * [zeros(1, z_f) (y1(m_p+2:M)) y1(1:m_p+1) zeros(1,z_b)]
x2 = ifft(y2, N)
x3 = ifft(y3, N)

Also, why zero-padding at the ends of the spectrum gives negatives values of interpolated data and positive of original sequence?

x2 =
2.0000 + 0i
 2.1303 + 0.0000i 
 4.0000 - 0.0000i 
 5.2490 + 0i 
 5.0000 + 0i
 3.6885 - 0.0000i 
 2.0000 + 0i
 0.8029 - 0.0000i
 1.0000 + 0.0000i
 2.3184 + 0i
 3.0000 - 0.0000i
 1.9939 + 0.0000i
 1.0000 + 0i
 2.7335 - 0.0000i
 7.0000 - 0.0000i
10.0992 + 0i
 9.0000 + 0.0000i 
 4.9842 +  0.0000i
x3 =
2.0000 + 0i
 -2.1303 - 0.0000i
  4.0000 - 0.0000i
 -5.2490 + 0i 
  5.0000 + 0i
 -3.6885 + 0.0000i
  2.0000 + 0i
 -0.8029 + 0.0000i
  1.0000 + 0.0000i
 -2.3184 + 0i
  3.0000 - 0.0000i
 -1.9939 - 0.0000i
  1.0000 + 0i
 -2.7335 + 0.0000i
  7.0000 - 0.0000i

-10.0992 + 0i
  9.0000 + 0.0000i
 -4.9842 - 0.0000i

Answer 1

I ... discovered that original sequence in interpolated data only when (N/M) = 2

Whatever you did, you did discover wrongly. Provided you zero-pad correctly, the original samples will be preserved whenever the ratio is an integer, i.e. $N/M \in \mathbb{I}$

If the ratio is not an integer, the new sample grid doesn't include the sampling times of the original samples, hence they cannot be preserved.

Also, why zero-padding at the ends of the spectrum gives negatives values of interpolated data and positive of original sequence?

Because you created a frequency shifted version of the correctly zero-padded spectrum. The shift is exactly half the sequence. This can be interpreted as modulating with a complex sine with a frequency of $N/2$, i.e.

$$x_{mod}[n] = e^{j2\pi\frac{N/2 \cdot n}{N}} = e^{j2\pi\frac{n}{2}} = e^{j\pi n} = (-1)^n $$