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By combining high-sensitivity LOFAR 150MHz, uGMRT 400MHz, GMRT 610MHz, andJVLA 5GHz data in the ELAIS-N1 field, we study the radio spectral properties ofradio-detected star-forming galaxies (SFGs) at observer-frame frequencies of150-5000MHz. We select ~3,500 SFGs that have both LOFAR 150MHz and GMRT 610MHzdetections by removing AGN from the two radio samples, and obtain a medianspectral index of $\alpha_{150}^{610}=-0.51\pm0.01$ with a scatter of$\sigma=0.2$. Due to the relatively lower sensitivity of uGMRT 400MHz data, weapply a flux cut of $S_{610}>300\mu$Jy and obtain the median spectral indicesof $\alpha_{150}^{385}=-0.42^{+0.03}_{-0.02}$,$\alpha_{385}^{610}=-0.44^{+0.03}_{-0.04}$, and$\alpha_{150}^{610}=-0.42^{+0.02}_{-0.01}$ for the sample of 258 SFGs that havedetections at these three radio frequencies. The JVLA 5GHz observations onlycover the central 0.1deg$^{2}$, where ~100 SFGs are selected, for which weobtain median $\alpha_{610}^{5000}=-1.14^{+0.04}_{-0.05}$,$\alpha_{385}^{5000}=-1.08^{+0.01}_{-0.02}$ and$\alpha_{150}^{5000}=-0.87\pm0.01$. Overall, the results show that the radiospectrum is flatter if we include a lower frequency dataset when measuring theradio spectral index at 150-5000MHz. We study the correlations between radiospectral index and physical properties of radio-selected SFGs and find that, onaverage, the radio spectrum slightly steepens with increasing stellar mass.However, we only find that the radio spectrum flattens with increasing opticaldepth at V-band at $\nu<\sim1$GHz. We suggest that spectral ageing due to theenergy loss of cosmic ray electrons and thermal free-free absorption could beamong the most likely physical mechanisms that drive the two correlationsrespectively. Both of these could be the physical causes of why the radiospectrum is flatter at low frequency than at high frequency.

Radio spectral properties at 150-5000MHz of star-forming galaxies in the ELAIS-N1 field